Regularity of bicyclic Graphs and their powers
Yairon Cid-Ruiz, Sepehr Jafari, Navid Nemati, Beatrice Picone
aa r X i v : . [ m a t h . A C ] O c t REGULARITY OF BICYCLIC GRAPHS AND THEIR POWERS
YAIRON CID-RUIZ, SEPEHR JAFARI, NAVID NEMATI, AND BEATRICE PICONE
Abstract.
Let I ( G ) be the edge ideal of a bicyclic graph. In this paper,we characterize the Castelnuovo-Mumford regularity of I ( G ) in terms of theinduced matching number of G . For the base case of this family of graphs,i.e. dumbbell graphs, we explicitly compute the induced matching number.Moreover, we prove that reg I ( G ) q = 2 q + reg I ( G ) −
2, for all q ≥
1, when G is a dumbbell graph with a connecting path having no more than two vertices. Introduction
Let I be a homogeneous ideal of the polynomial ring R = K [ x , . . . , x r ]. TheCastelnuovo-Mumford regularity of I , denoted by reg ( I ), has been an interestingand active research topic for the past decades. There exists a vast literature onthe study of the reg ( I ). One of the most important results on the behavior ofthe regularity of powers of ideals was given independently by Cutkosky, Herzog,and Trung in [9], and by Kodiyalam in [23]. In both papers, it is proved thatfor all q ≥ q , the regularity of powers of I is asymptotically a linear functionreg ( I q ) = dq + b , where q is the so-called stabilizing index, and b is the so-calledconstant. The value of d in the above formula is well understood. For example, d is equal to the degree of the generators of I when I is equigenerated. However,their method does not give precise information on q and b .Since then, many researchers have tried to compute q and b for special familiesof ideals. The most simple case, yet interesting, is when I is the edge ideal of afinite simple graph. Let G = ( V ( G ) , E ( G )) denote a finite simple undirectedgraph. Let R be the polynomial ring K [ x i | x i ∈ V ( G )] where K is any field. Theedge ideal I ( G ) of G is the ideal I ( G ) = ( x i x j | { x i , x j } ∈ E ( G )) . Several authors have settled the problem of determining the stabilizing index andthe constant for special families of graphs. Banerjee proved that reg I ( G ) q = 2 q ,for all q ≥
2, when G is a gap-free and cricket-free graph (see [4]). Moghimian,Fakhari, and Yassemi answered the question for the family of whiskered graphs(see [25]). Beyarslan, H`a, and Trung settled the problem for the family of forestsand cycles (see [6]). Their results were expanded to the family of unicyclic graphsby Alilooee, Beyarslan, and Selvaraja (see [1]). Moreover, Alilooee and Baner-jee determined the stabilizing index and the constant for the family of bipartitegraphs with regularity equal to three (see [2]). Jayanthan and Selvaraja settledthe problem for the family of very well-covered graphs (see [20]). Recently, Ereyproved that if G is a gap-free and diamond-free graph, then reg I ( G ) q = 2 q forall q ≥ Mathematics Subject Classification.
Key words and phrases. bicyclic graphs, edge ideals, regularity, induced matching number, Lozintransformation, even-connection.The first named author was funded by the European Union’s Horizon 2020 research and innova-tion programme under the Marie Sk lodowska-Curie grant agreement No. 675789. see [22], [16], [8], [18], [3], [27] and [26] for more information on this topic. Thepurpose of this paper is to extend the results of [1] to the family of bicyclic graphs(i.e. a graph with exactly two cycles).The base case of the family of bicyclic graphs is that of dumbbell graphs. Adumbbell graph C n · P l · C m is a graph consisting of two cycles C n and C m connectedwith a path P l , where n , m , and l are the number of vertices (see Example 2.1).For convenience of notation, we define the following function ξ ( n ) = ( n ≡ , , n ≡ . Here, we describe the basic outline and main results of this paper.In Section 1, we fix some notations and recall known results which are crucialto our approach.In Section 2, we use combinatorial techniques to compute the induced matchingnumber of a dumbbell graph. Then, applying inductive methods, we study theregularity of the edge ideals of dumbbell graphs. For a dumbbell graph C n · P l · C m ,we will always assume that n mod 3 ≤ m mod 3. The cases n ≡ m ≡ , n ≡ , m ≡ Theorem A (Theorem 2.4) . Let n, m ≥ and l ≥ , then ν ( C n · P l · C m ) = j n k + j m k + j l − ξ ( n ) − ξ ( m ) + 13 k . Theorem B (Theorem 2.6) . Let m, n ≥ and l ≥ ,(i) if l ≡ , , then reg I ( C n · P l · C m ) = ( ν ( C n · P l · C m ) + 2 if n, m ≡ ,ν ( C n · P l · C m ) + 1 otherwise; (ii) if l ≡ , then reg I ( C n · P l · C m ) = ( ν ( C n · P l · C m ) + 2 if n ≡ , , m ≡ ν ( C n · P l · C m ) + 1 otherwise . In Section 3, for an arbitrary bicyclic graph G , we give a combinatorial char-acterization of reg I ( G ) in terms of the induced matching number ν ( G ). Theorem C (Theorem 3.2) . Let G be a bicyclic graph with dumbbell C n · P l · C m .The following statements hold.(I) Let n, m ≡ , , then reg I ( G ) = ν ( G ) + 1 .(II) Let n ≡ , and m ≡ , then ν ( G ) + 1 ≤ reg I ( G ) ≤ ν ( G ) + 2 , and reg I ( G ) = ν ( G ) + 2 if and only if ν ( G ) = ν ( G \ Γ G ( C m )) .(III) Let n, m ≡ and l ≥ , then ν ( G ) + 1 ≤ reg I ( G ) ≤ ν ( G ) + 3 .Moreover:(i) reg I ( G ) = ν ( G ) + 3 if and only if ν ( G \ Γ G ( C n ∪ C m )) = ν ( G ) .(ii) reg I ( G ) = ν ( G ) + 1 if and only if the following conditions hold:(a) ν ( G ) − ν ( G \ Γ G ( C n ∪ C m )) > ; EGULARITY OF BICYCLIC GRAPHS AND THEIR POWERS 3 (b) ν ( G ) > ν ( G \ Γ G ( C n )) ;(c) ν ( G ) > ν ( G \ Γ G ( C m )) .(IV) Let n, m ≡ and l ≤ , then ν ( G ) + 1 ≤ reg I ( G ) ≤ ν ( G ) + 2 . If x is a vertex on P l and L x ( G ) is the Lozin transformation of G with respectto x , then reg I ( G ) = ν ( G ) + 1 if and only if the following conditions aresatisfied:(a) ν ( L x ( G )) − ν ( L x ( G ) \ Γ L x ( G ) ( C n ∪ C m )) > ;(b) ν ( L x ( G )) > ν ( L x ( G ) \ Γ L x ( G ) ( C n )) ;(c) ν ( L x ( G )) > ν ( L x ( G ) \ Γ L x ( G ) ( C m )) . In Section 4, we investigate the asymptotic behavior of regularity of powersof I ( C n · P l · C m ) when l ≤
2. The approach takes advantage of the notion ofeven-connectedness and the relations between the induced matching number ofgraphs and the regularity of the edge ideal.
Theorem D (Theorem 4.6) . Let C n · P l · C m with l ≤ , then reg I ( C n · P l · C m ) q = 2 q + reg I ( C n · P l · C m ) − for any q ≥ . For the case l ≥
3, there are immediate examples for which the above theoremdoes not hold (see Remark 4.8).1.
Preliminaries
Let R = K [ x , . . . , x r ] be the standard graded polynomial ring over a field K and let m = ( x , . . . , x r ) be its maximal homogeneous ideal. For a graded R -module M , one can define the Castelnuovo-Mumford regularity in different terms.We recall the definition of the regularity of an R -module M by the minimal freeresolution M . The minimal graded free resolution of M is an exact sequence ofthe form 0 → F p → F p − → · · · → F → M → , where each F i is a graded free R -module of the form F i = L j ∈ N R ( − j ) β i,j ( M ) , each ϕ i : F i → F i − , with F − := M , is a graded homomorphism of degree zerosuch that ϕ i +1 ( F i +1 ) ⊆ m F i for all i ≥
0. The numbers β i,j ( M ) are importantinvariants, known as the graded Betti numbers of M . In particular, the number β i = P j ∈ N β i,j ( M ) is called the i-th Betti number of M and β i,j ( M ) is the i-th Bettinumber of M of degree j . Note that the minimal free resolution of M is uniqueup to isomorphism, hence the graded Betti numbers are uniquely determined. Definition 1.1.
Let M be a finitely generated graded R -module. The regularityof M is given by reg ( M ) = max { j − i | β i,j ( M ) = 0 } . Remark 1.2.
Note that, if I is a graded ideal of R , then reg ( R/I ) = reg ( I ) − . Let G = ( V, E ) be a graph with vertex set V = { v , . . . , v l } . Here, we recallsome classes of graphs that we need for this study. Definition 1.3.
Let G = ( V, E ) be a graph.(i) G is called a path with l vertices, denoted by P l , if V = { v , . . . , v l } and { v i , v i +1 } ∈ E for all ≤ i ≤ l − . YAIRON CID-RUIZ, SEPEHR JAFARI, NAVID NEMATI, AND BEATRICE PICONE (ii) G is called a cycle with n vertices, denoted by C n , if V = { v , . . . , v n } and { v i , v i +1 } ∈ E for all ≤ i ≤ n − and { v n , v } ∈ E .(iii) G is called a dumbbell graph if G contains two cycles C n and C m joined bya path P l of l vertices. We denote it by C n · P l · C m . (See Example 2.1) For a vertex u in a graph G = ( V, E ), let N G ( u ) = { v ∈ V |{ u, v } ∈ E } be theset of neighbors of u , and set N G [ u ] := N G ( u ) ∪ { u } . An edge e is incident toa vertex u if u ∈ e . The degree of a vertex u ∈ V , denoted by deg G ( u ), is thenumber of edges incident to u . When there is no confusion, we will omit G andwrite N ( u ) , N [ u ] and deg( u ). For an edge e in a graph G = ( V, E ), we define G \ e to be the subgraph of G obtained by deleting e from E (but the verticesare remained). For a subset W ⊆ V of the vertices in G , we define G \ W tobe the subgraph of G deleting the vertices of W and their incident edges. When W = { u } consists of a single vertex, we write G \ u instead of G \ { u } . For anedge e = { u, v } ∈ E , let N G [ e ] = N G [ u ] ∪ N G [ v ] and define G e to be the inducedsubgraph of G over the vertex set V \ N G [ e ].One can think of the vertices of G = ( V, E ) as the variables of the polynomialring R = K [ x , . . . , x r ] for convenience. Similarly, the edges of G can be consideredas square free monomials of degree two. By abuse of notation, we use e to referto both the edge e = { x i , x j } and the monomial e = x i x j ∈ I ( G ).Let G = ( V, E ) be a graph and W ⊆ V . The induced subgraph of G on W ,denoted by G [ W ], is the graph with vertex set W and edge set { e ∈ E | e ⊆ W } . Definition 1.4.
Let G = ( V, E ) be a graph.A collection C of edges of G is called a matching if the edges in C are pairwisedisjoint. The maximum size of a matching in G is called its matching number,which is denoted by match( G ) .A collection C of edges of G is called an induced matching if C is a matching,and C consists of all edges of the induced subgraph G (cid:2) S e ∈ C e (cid:3) of G . The maximumsize of an induced matching in G is called its induced matching number and it isdenoted by ν ( G ) . Remark 1.5. ( [6, Remark 2.12] ) Let P l be a path of l vertices, then we have ν ( P l ) = ⌊ l + 13 ⌋ Remark 1.6. ( [6, Remark 2.13] ) Let C n be a cycle of n vertices, then we have ν ( C n ) = ⌊ n ⌋ . Depending on r = n mod 3 we can assume the following:(i) when r = 0 , there exists a maximal induced matching of C n that does notcontain the edges x x and x x n ;(ii) when r = 1 , there exists a maximal induced matching of C n that does notcontain the edges x x , x x n and x n − x n ;(iii) when r = 2 , there exists a maximal induced matching of C n that does notcontain the edges x x , x x , x x n and x n − x n . Theorem 1.7. [15, Lemma 3.1, Theorems 3.4 and 3.5]
Let G = ( V, E ) be a graph.(i) If H is an induced subgraph of G , then reg I ( H ) ≤ reg I ( G ); (ii) Let x ∈ V , then reg I ( G ) ≤ max { reg I ( G \ x ) , reg I ( G \ N [ x ]) + 1 } ; EGULARITY OF BICYCLIC GRAPHS AND THEIR POWERS 5 (iii) Let e ∈ E , then reg I ( G ) ≤ max { , reg I ( G \ e ) , reg I ( G e ) + 1 } . Now we recall the concept of even-connection introduced by Banerjee in [4].
Definition 1.8 ([4]) . Let G = ( V, E ) be a graph with edge ideal I = I ( G ) . Twovertices x i and x j in G are called even-connected with respect to an s -fold product M = e · · · e s , where e , . . . , e s are edges in G , if there is a path p , . . . , p l +1 , forsome l ≥ , in G such that the following conditions hold:(i) p = x i and p l +1 = x j ;(ii) for all ≤ j ≤ l − , { p j +1 , p j +2 } = e i for some i ;(iii) for all i , (cid:12)(cid:12) { j | { p j +1 , p j +2 } = e i } (cid:12)(cid:12) ≤ (cid:12)(cid:12) { t | e t = e i } (cid:12)(cid:12) . Theorem 1.9. [4, Theorems 6.1 and 6.5]
Let M = e e · · · e s be a minimal gen-erator of I s . Then ( I s +1 : M ) is minimally generated by monomials of degree ,and uv ( u and v may be the same) is a minimal generator of ( I s +1 : M ) if andonly if either { u, v } ∈ E or u and v are even-connected with respect to M . Remark 1.10. [4, Lemma 6.11]
Let ( I s +1 : M ) pol be the polarization of the ideal ( I s +1 : M ) (see e.g. [17, § ). From the previous theorem we can construct agraph G ′ whose edge ideal is given by ( I s +1 : M ) pol . The new graph G ′ is given by:(i) All the vertices and edges of G .(ii) Any two vertices u, v , u = v that are even-connected with respect to M areconnected by an edge in G ′ .(iii) For every vertex u which is even-connected to itself with respect to M , thereis a new vertex u ′ which is connected to u by an edge and not connected toany other vertex (so uu ′ is a whisker). Theorem 1.11. [4, Theorem 5.2]
Let G be a graph and { m , . . . , m r } be the setof minimal monomial generators of I ( G ) q for all q ≥ , then reg I ( G ) q +1 ≤ max { reg ( I ( G ) q : m l ) + 2 q, ≤ l ≤ r, reg I ( G ) q } . Here by, we recall a result by Kalai and Meshulam on the regularity of monomialideals.
Theorem 1.12. [21]
Let I , . . . , I s be monomial ideals in R , then reg R . s X i =1 I i ! ≤ s X i =1 reg ( R/I i ) . The regularity of the edge ideal of a forest was first computed by Zheng in[28, Theorem 2.18].
Theorem 1.13. [28, Theorem 2.18]
Let G be a forest, then reg I ( G ) = ν ( G ) + 1 . In [22] Katzman first noticed that the previous equality is a lower bound forgeneral graphs.
Theorem 1.14. [22, Corollary 1.2]
Let G be a graph, then reg I ( G ) ≥ ν ( G ) + 1 . The decycling number of a graph is an important combinatorial invariant whichcan be used to obtain an upper bound for the regularity of the edge ideal of agraph.
YAIRON CID-RUIZ, SEPEHR JAFARI, NAVID NEMATI, AND BEATRICE PICONE
Definition 1.15.
For a graph G and D ⊂ V ( G ) , if G \ D is acyclic, i.e. containsno induced cycle, then D is said to be a decycling set of G . The size of a smallestdecycling set of G is called the decycling number of G and denoted by ∇ ( G ) . Theorem 1.16. [7, Theorem 4.11]
Let G be a graph, then reg I ( G ) ≤ ν ( G ) + ∇ ( G ) + 1 . In [6] Beyarslan, H`a and Trung provided a formula for the regularity of thepowers of edge ideals of forests and cycles in terms of the induced matchingnumber.
Theorem 1.17. [6, Theorem 4.7]
Let G be a forest, then reg I ( G ) q = 2 q + ν ( G ) − . for all q ≥ . Theorem 1.18. [6, Theorem 5.2] . Let C n be a cycle with n vertices, then reg I ( C n ) = ( ν ( C n ) + 1 if n ≡ , ,ν ( C n ) + 2 if n ≡ , where ν ( C n ) = ⌊ n ⌋ denote the induced matching number of C n . Moreover, reg I ( C n ) q = 2 q + ν ( C n ) − . and for all q ≥ . In addition, the authors of [6] provided a lower bound for the regularity ofthe powers of the edge ideal of an arbitrary graph, and an upper bound for theregularity of the edge ideal of a graph containing a Hamiltonian path.
Theorem 1.19. [6, Theorem 4.5]
Let G be a graph and let ν ( G ) denote its inducedmatching number. Then, for all q ≥ , we have reg I ( G ) q ≥ q + ν ( G ) − Theorem 1.20. [6, Theorem 3.1]
Let G be a graph on n vertices. Assume G contains a Hamiltonian path, then reg I ( G ) ≤ ⌊ n + 13 ⌋ + 12. Regularity and induced matching number of a dumbbell graph
In this section we compute the induced matching number of a dumbbell graphand the regularity of its edge ideal. Recall that C n · P l · C m denotes the graphconstructed by joining two cycles C n and C m via a path P l . In this section, we de-note the vertices of C n , C m and P l by { x , . . . , x n } , { y , . . . , y m } and { z , . . . , z l } ,respectively. We make the identifications x = z and y = z l . Example 2.1.
Two base cases when l = 2 and l = 1 are the following: EGULARITY OF BICYCLIC GRAPHS AND THEIR POWERS 7 x x x = z y = z y y x x x = y = z y y y Figure 1.
The graphs C · P · C and C · P · C . Notation 2.2.
Let ξ be the function defined as below ξ ( n ) = ( if n ≡ , , if n ≡ . Let C n · P l be the graph given by connecting the path P l to the cycle C n . Forinstance, the graph C · P can be illustrated as the following: x x x = z z z Proposition 2.3.
Let n ≥ and l ≥ , then ν ( C n · P l ) = j n k + j l − ξ ( n ) + 13 k . Proof.
Case 1: From Remark 1.6, in the case n ≡ x are not chosen in a maximal induced matching of C n . Then, we can choose theedges in P l without any constraint coming from the maximal induced matchingchosen in C n , and so we have ν ( C n · P l ) = ⌊ n ⌋ + ⌊ l +13 ⌋ .Case 2: It remain to consider the case ξ ( n ) = 1, i.e., n ≡ , M be an induced matching of maximal size in G . We analyze separately the twocases of whether z z (the edge adjacent to the cycle C n ) is in M or not.Suppose z z is not an edge of M . Then M can be considered as the union ofa maximal matching of C n as introduced in Remark 1.6 and a maximal matchingof the path P l \ z . Thus |M| = ν ( C n ) + ν ( P l − ) = ⌊ n ⌋ + ⌊ ( l − ⌋ .If z z ∈ M , then none of the edges incident to the vertices in N C n [ x ] = { x , x , x n } are in M | C n := { e ∈ M | e ∈ C n } . Hence |M | C n | = ν ( P n − ),and since n ≡ , |M | C n | = ⌊ n − ⌋ = ⌊ n ⌋ −
1. From z z ∈ M we get |M | P l | = ν ( P l ) = ⌊ l +13 ⌋ . So, by joining both computations weget |M| = ⌊ n ⌋ − ⌊ l +13 ⌋ = ⌊ n ⌋ + ⌊ l − ⌋ . Therefore, we obtain that ν ( C n · P l ) = ⌊ n ⌋ + ⌊ ( l − ⌋ . (cid:3) Theorem 2.4.
Let n, m ≥ and l ≥ , then ν ( C n · P l · C m ) = j n k + j m k + j l − ξ ( n ) − ξ ( m ) + 13 k . YAIRON CID-RUIZ, SEPEHR JAFARI, NAVID NEMATI, AND BEATRICE PICONE
Proof.
We use the same argument as in Proposition 2.3. By Remark 1.6 we havethat when either n ≡ m ≡ C n or in C m does not affect the way we choose edges in the path P l .In the case n ≡ , C n , which is the same as sayingthat we are not going to use one extreme vertex of the path P l . Similarly, when m ≡ , (cid:3) The aim of the rest of this section is to explicitly compute the regularity of I ( C n · P l · C m ) in term of the induced matching number. We divide it into threesubsections depending on the value of l mod 3. The base of our computations isgiven by the following proposition. Proposition 2.5.
Let n, m ≥ and l ≥ , then reg I ( C n · P l · C m ) − ν ( C n · P l · C m ) = reg I ( C n · P l +3 · C m ) − ν ( C n · P l +3 · C m ) . Proof.
From the formula obtained in Theorem 2.4 or [24, Lemma 1], we have theequality ν ( C n · P l +3 · C m ) = ν ( C n · P l · C m ) + 1 . We can apply the Lozin transformation (see e.g. [24], [7]) to any of the verticesin the bridge P l , then from [7, Theorem 1.1] we havereg I ( C n · P l +3 · C m ) = reg I ( C n · P l · C m ) + 1 . Thus, the statement of the proposition follows by subtracting these equalities. (cid:3)
From the previous proposition, it follows that we only need to consider thecases l = 1, l = 2 and l = 3. We treat each case in a separate subsection. In thefollowing theorem we compute the regularity of the edge ideal of the dumbbell C n · P l · C m . Theorem 2.6.
Let m, n ≥ and l ≥ , then(i) if l ≡ , , then reg I ( C n · P l · C m ) = ( ν ( C n · P l · C m ) + 2 if n, m ≡ ,ν ( C n · P l · C m ) + 1 otherwise; (ii) if l ≡ , then reg I ( C n · P l · C m ) = ( ν ( C n · P l · C m ) + 2 n ≡ , , m ≡ ν ( C n · P l · C m ) + 1 otherwise . Proof.
Follows from Proposition 2.5, and Theorem 2.8, Theorem 2.14, and Theorem 2.16. (cid:3)
The basic approach in the next three subsections is to obtain lower and upperbounds that coincide.2.1.
The case l = 1 . Throughout this subsection, we consider the dumbbell graph C n · P · C m . Proposition 2.7.
Let n, m ≥ , then reg I ( C n · P · C m ) ≤ max nj n k + j m k + 1 , j n − k + j m − k + 2 o . Moreover, reg I ( C n · P · C m ) is equal to one of these terms. EGULARITY OF BICYCLIC GRAPHS AND THEIR POWERS 9
Proof.
We use [10, Lemma 3.2], that gives an improved version of the exact se-quence coming from deleting the vertex z . We havereg I ( C n · P · C m ) ∈ n reg I (cid:0) ( C n · P · C m ) \ z (cid:1) , reg I (cid:0) ( C n · P · C m ) \ N [ z ] (cid:1) +1 o . Since ( C n · P · C m ) \ z = P n − ∪ P m − and ( C n · P · C m ) \ N [ z ] = P n − ∪ P m − ,we get the result by applying Theorem 1.13. (cid:3) Theorem 2.8.
Let n, m ≥ , then reg I ( C n · P · C m ) = ( ν ( C n · P · C m ) + 2 if n ≡ , m ≡ ν ( C n · P · C m ) + 1 otherwise . Proof.
Suppose n ≡ m ≡ ⌊ k − ⌋ = ⌊ k ⌋ when k ≡ {⌊ n ⌋ + ⌊ m ⌋ + 1 , ⌊ n − ⌋ + ⌊ m − ⌋ + 2 } = ⌊ n ⌋ + ⌊ m ⌋ + 2 . Thus Proposition 2.7 yields(1) reg I ( C n · P · C m ) ≤ ⌊ n ⌋ + ⌊ m ⌋ + 2 . Consider the induced subgraph H = ( C n · P · C m ) \ { x n } where x n is in C n andit is incident to x (e.g. see x in Example 2.1). In fact, H is the graph given byjoining C m and a path P n − , that is, H = C m · P n − . Now from Proposition 2.3,we have that ν ( H ) = ⌊ n ⌋ + ⌊ m ⌋ . By Theorem 1.7 ( i ), we get reg I ( C n · P · C m ) ≥ reg I ( H ). From [1, Theorem 1.1], we have reg I ( H ) = ν ( H ) + 2. Therefore, theequality holds in (1). The proof of this part is complete since Theorem 2.4 yields ν ( C n · P · C m ) = ⌊ n ⌋ + ⌊ m ⌋ .For any case distinct to n ≡ m ≡ {⌊ n ⌋ + ⌊ m ⌋ + 1 , ⌊ n − ⌋ + ⌊ m − ⌋ + 2 } = ⌊ n ⌋ + ⌊ m ⌋ + 1 . Therefore, from Proposition 2.7, we have(2) reg I ( C n · P · C m ) ≤ ⌊ n ⌋ + ⌊ m ⌋ + 1 . From Theorem 2.4, we have ν ( C n · P · C m ) = ⌊ n ⌋ + ⌊ m ⌋ . Moreover, Theorem 1.14gives reg I ( C n · P · C m ) ≥ ν ( C n · P · C m ) + 1. Thus, the equality in (2) holds.Therefore the proof is complete. (cid:3) The case l = 2 . Throughout this subsection, we consider the dumbbell graph C n · P · C m . Remark 2.9.
The regularity of I ( C n ) is given in Theorem 1.18. For simplicityof notation, we use the equivalent formula reg I ( C n ) = ⌊ n − ⌋ + 2 . Proposition 2.10.
Let n, m ≥ , then (3) ν ( C n · P · C m ) ≤ reg ( RI ( C n · P · C m ) ) ≤ ⌊ n − ⌋ + ⌊ m − ⌋ + 2 . Proof.
We only need to prove the inequality on the right since the lower boundis given due to Theorem 1.14 and reg ( J ) − RJ ) for any ideal of J ⊂ R .In the original graph C n · P · C m we shall remove the edge that connects the twocycles C n and C m . . The set of vertices of C n and C m are given respectivelyby { x , . . . , x n } and { y , . . . , y m } , and we assume that the edge e = x y is the bridge between the two cycles. Also, we denote by C n ∪ C m the resulting graphgiven as the disjoint union of the two cycles C n and C m . Thus Theorem 1.7( iii )yields the inequalityreg (cid:18) RI ( C n · P · C m ) (cid:19) ≤ max n reg (cid:18) RI ( C n ∪ C m ) : e (cid:19) + 1 , reg (cid:18) RI ( C n ∪ C m ) (cid:19)o . From [19, Lemma 3.2] we have that the regularity of the two disjoint cycles C n ∪ C m is given byreg (cid:18) RI ( C n ∪ C m ) (cid:19) = reg (cid:18) RI ( C n ) (cid:19) + reg (cid:18) RI ( C m ) (cid:19) , and using Remark 2.9 we get the equalityreg (cid:18) RI ( C n ∪ C m ) (cid:19) = j n − k + j m − k + 2 . Consider the graph H = { x , x n } ∪ P n − ∪ { y , y m } ∪ P m − , where { x , x n } and { y , y m } are incident vertices of graph C n · P · C m to x and y respectively(see Example 2.1). Moreover, P n − is the path with vertices x , . . . , x n − and P m − is the path with vertices y , . . . , y m − . It is easy to see that reg I ( H ) =reg I ( C n ∪ C m ) : e . Hence from Remark 1.5, Theorem 1.12 and again [19, Lemma3.2] we get reg (cid:18) RI ( C n ∪ C m ) : e (cid:19) + 1 = j n − k + j m − k + 1 , This proves the proposition. (cid:3)
As a result of the previous proposition, we can prove the following corollary.
Corollary 2.11. If n ≡ , and m ≡ , , then reg (cid:18) RI ( C n · P · C m ) (cid:19) = ν ( C n · P · C m ) = j n k + j m k Proof.
We note that ⌊ k ⌋ = ⌊ k − ⌋ + 1 when k ≡ , (cid:3) Now we have only three more cases left to deal with, i.e., the case n ≡ , m ≡ n ≡ , m ≡ n ≡ , m ≡ Lemma 2.12. If n ≡ and m ≡ , then reg (cid:18) RI ( C n · P · C m ) (cid:19) = ν ( C n · P · C m ) = j n k + j m k + 1 . Proof.
We shall divide the graph into three subgraphs H , H and H . We make H = C n \ { x } and H = C m \ { y } . The subgraph H is defined by takingthe bridge e = x y and the neighboring vertices { x , x n , y , y m } , i.e. the graphbelow. EGULARITY OF BICYCLIC GRAPHS AND THEIR POWERS 11 x y y y m x n x Using this decomposition and Theorem 1.12 we get the inequalityreg
R/I ( C n · P · C m ) ≤ reg ( R/I ( H )) + reg ( R/I ( H )) + reg ( R/I ( H )) , then have that H and H are paths of length n − m − R/I ( C n · P · C m ) ≤ j n k + j m k + 1 . Finally, in the present case n ≡ m ≡ ν ( C n · P · C m ) = ⌊ n ⌋ + ⌊ m ⌋ + 1, and the proof follows from Theorem 1.14. (cid:3) Lemma 2.13. If n ≡ , and m ≡ , then reg (cid:18) RI ( C n · P · C m ) (cid:19) = ν ( C n · P · C m ) + 1 = j n k + j m k + 1 . Proof.
In this case we will delete the vertex x from the cycle C n . We have that H = ( C n · P · C m ) \ { x } is an induced subgraph of C n · P · C m which is given asthe disjoint union of a path of length n − m , i.e. H = P n − ∪ C m .From Theorem 1.7( i ) we get thatreg ( R/I ( C n · P · C m )) ≥ reg ( R/I ( H )) = j n k + j m k + 1 . It follows from Proposition 2.10 and the fact that ⌊ k/ ⌋ = ⌊ ( k − / ⌋ + 1 when k ≡ , R/I ( C n · P · C m ) = j n k + j m k + 1 . (cid:3) Theorem 2.14.
Let n, m ≥ , then reg I ( C n · P · C m ) = ( ν ( C n · P · C m ) + 2 if n ≡ , , m ≡ ν ( C n · P · C m ) + 1 otherwise . Proof.
It follows by Corollary 2.11, Lemma 2.12 and Lemma 2.13. (cid:3)
The case l = 3 . Throughout this subsection, we consider the dumbbell graph C n · P · C m . Proposition 2.15.
Let n, m ≥ , then(i) reg I ( C n · P · C m ) ≤ ν ( C n · P · C m ) + 2 , if n, m ≡ ;(ii) reg I ( C n · P · C m ) = ν ( C n · P · C m ) + 1 , otherwise.Proof. Let E ( P ) = { e, e ′ } be the set of the edges of P , where e = z z and e ′ = z z are connected to C n and C m , respectively. Since reg ( I ( C n ∪ ( e ′ · C m )) : e ) =reg ( I ( P n − ∪ P m − )), then Theorem 1.7( iii ) yields the inequalityreg (cid:18) RI ( C n · P · C m ) (cid:19) ≤ max n reg (cid:18) RI ( P n − ∪ P m − ) (cid:19) +1 , reg (cid:18) RI ( C n ∪ ( e ′ · C m ) (cid:19)o . From Proposition 2.3 and [1, Lemma 3.2] follows that reg ( I ( e ′ · C m )) = ⌊ m ⌋ + ⌊ − ξ ( m )3 ⌋ + 1. Thus, using Remark 2.9, [19, Lemma 3.2] and Theorem 1.13, weget reg (cid:16) RI ( C n · P · C m ) (cid:17) ≤ max nj n − k + j m k + 1 , j n − k + 1 + j m k + j − ξ ( m )3 ko . On the other hand, from Theorem 2.4 we have that ν ( C n · P · C m ) = ⌊ n ⌋ + ⌊ m ⌋ + ⌊ − ξ ( n ) − ξ ( m )3 ⌋ . Therefore, we can check that reg (cid:16) RI ( C n · P · C m ) (cid:17) ≤ ν ( C n · P · C m ) + 1 when n, m ≡ (cid:16) RI ( C n · P · C m ) (cid:17) = ν ( C n · P · C m )in all the remaining cases. (cid:3) Theorem 2.16.
Let n, m ≥ , then reg I ( C n · P · C m ) = ( ν ( C n · P · C m ) + 2 if n, m ≡ ,ν ( C n · P · C m ) + 1 otherwise . Proof.
Using Proposition 2.15, then we only need to prove that reg I ( C n · P · C m ) ≥ ν ( C n · P · C m ) + 2 in the case n, m ≡ n, m ≡ z be the middle vertex of C n · P · C m . By deleting z we see that H = ( C n · P · C m ) \ z = C n ∪ C m is an induced subgraph of C n · P · C m . FromTheorem 1.18 and [19, Lemma 3.2], we have thatreg I ( H ) = reg I ( C n ) + reg I ( C m ) − ν ( C n ) + ν ( C m ) + 3 . Since ν ( C n · P · C m ) = ν ( C n ) + ν ( C m ) + 1, then using Theorem 1.7( i ) we getreg I ( C n · P · C m ) ≥ reg I ( H ) = ν ( C n · P · C m ) + 2 . (cid:3) Combinatorial characterization of reg ( I ( G )) in terms of ν ( G )Let G be a general bicyclic graph, then its decycling number is smaller or equalthan 2, and so from Theorem 1.14 and Theorem 1.16, we get ν ( G ) + 1 ≤ reg I ( G ) ≤ ν ( G ) + 3 . Example 3.1.
The following graph G x x x x x z y z z y y y y has regularity reg I ( G ) = 6 and induced matching number ν ( G ) = 3 . In this section, we give a combinatorial characterization of the bicyclic graphswith regularity ν ( G ) + 1, ν ( G ) + 2 and ν ( G ) + 3. For the rest of the paper, we shalluse the term “dumbbell” of the bicyclic graph G , and it denotes the unique subgraph of G of the form C n · P l · C m . The theorem below contains the characterizationthat we found. Theorem 3.2.
Let G be a bicyclic graph with dumbbell C n · P l · C m . The followingstatements hold.(I) Let n, m ≡ , , then reg I ( G ) = ν ( G ) + 1 . EGULARITY OF BICYCLIC GRAPHS AND THEIR POWERS 13 (II) Let n ≡ , and m ≡ , then ν ( G ) + 1 ≤ reg I ( G ) ≤ ν ( G ) + 2 , and reg I ( G ) = ν ( G ) + 2 if and only if ν ( G ) = ν ( G \ Γ G ( C m )) .(III) Let n, m ≡ and l ≥ , then ν ( G ) + 1 ≤ reg I ( G ) ≤ ν ( G ) + 3 .Moreover:(i) reg I ( G ) = ν ( G ) + 3 if and only if ν ( G \ Γ G ( C n ∪ C m )) = ν ( G ) .(ii) reg I ( G ) = ν ( G ) + 1 if and only if the following conditions hold:(a) ν ( G ) − ν ( G \ Γ G ( C n ∪ C m )) > ;(b) ν ( G ) > ν ( G \ Γ G ( C n )) ;(c) ν ( G ) > ν ( G \ Γ G ( C m )) .(IV) Let n, m ≡ and l ≤ , then ν ( G ) + 1 ≤ reg I ( G ) ≤ ν ( G ) + 2 . If x is an edge on P l and L x ( G ) be the Lozin transformation of G with respectto x , then reg I ( G ) = ν ( G ) + 1 if and only if the following conditions aresatisfied:(a) ν ( L x ( G )) − ν ( L x ( G ) \ Γ L x ( G ) ( C n ∪ C m )) > ;(b) ν ( L x ( G )) > ν ( L x ( G ) \ Γ L x ( G ) ( C n )) ;(c) ν ( L x ( G )) > ν ( L x ( G ) \ Γ L x ( G ) ( C m )) .Proof. Statement (I) follows from Proposition 3.4. In Theorem 3.13, (II) is proved.By Theorem 3.18 and Theorem 3.23, we get (III) . Finally, from Corollary 3.24,we obtain (IV) . (cid:3) The following simple remark will be crucial in our treatment.
Remark 3.3. [1, Observation 2.1]
Let G be a graph with a leaf y and its uniqueneighbor x , say e = { x, y } . If { e , . . . , e s } is an induced matching in G \ N [ x ] , then { e , . . . , e s , e } is an induced matching in G . So we have ν ( G \ N [ x ]) + 1 ≤ ν ( G ) . Proposition 3.4.
Let G be a bicyclic graph with dumbbell C n · P l · C m . Thefollowing statements hold.(i) When n, m ≡ , , we have reg I ( G ) = ν ( G ) + 1 .(ii) When n ≡ , and m ≡ , we have reg I ( G ) ≤ ν ( G ) + 2 .(iii) When l ≤ , we have reg I ( G ) ≤ ν ( G ) + 2 .Proof. ( i ) Again, it is enough to prove the upper bound reg I ( G ) ≤ ν ( G ) + 1. Let E ′ be the set of edges E ′ = E ( G ) \ E ( C n · P l · C m ). We proceed by induction onthe cardinality of E ′ . If | E ′ | = 0 then the statement follows from Theorem 2.6,so we assume | E ′ | >
0. There exists a leaf y in G such that N [ y ] = { x } . Let G ′ = G \ x and G ′′ = G \ N [ x ], then by Theorem 1.7 we havereg I ( G ) ≤ max { reg I ( G ′ ) , reg I ( G ′′ ) + 1 } . The graphs G ′ and G ′′ can be either bicyclic graphs with the same dumbbell C n · P l · C m , or the disjoint union of two unicyclic graphs with cycles C n and C m , or unicyclic graphs with a cycle C r ( r = n or r = m ) of the type r ≡ , I ( G ′ ) = ν ( G ′ )+ 1 and reg I ( G ′′ ) = ν ( G ′′ )+ 1. Since we have ν ( G ′ ) ≤ ν ( G ) and ν ( G ′′ )+ 1 ≤ ν ( G ) (by Remark 3.3), then we obtain the required inequality.( ii ) and ( iii ) follow by the same inductive argument, only changing the fact that G ′ and G ′′ could be unicyclic graphs with cycle C r of the type r ≡ (cid:3) Remark 3.5.
The inductive process of the previous proposition cannot conclude reg I ( G ) ≤ ν ( G ) + 2 in the case l ≥ . Here we may encounter two disjointsubgraphs G and G with reg I ( G i ) = ν ( G i ) + 2 , which implies reg I ( G ∪ G ) = ν ( G ∪ G ) + 3 . This is exactly the case of Example 3.1.An alternative proof of the inequality reg I ( G ) ≤ ν ( G ) + 3 can be given by usingthe same inductive technique of Proposition 3.4. For the rest of the paper we shall use the following notation.
Notation 3.6.
Let G be a graph, H ⊂ G be a subgraph, and v and w be verticesof G . Then, we assume the following:(i) d ( v, w ) denotes the length (i.e., the number of edges) of a minimal pathbetween v and w . In particular, d ( v, v ) = 0 .(ii) d ( v, H ) denotes the minimal distance from the vertex v to the subgraph H ,that is d ( v, H ) = min { d ( v, w ) | w ∈ H } . In particular, d ( v, H ) = 0 if and only if v ∈ H .(iii) Let H ′ ⊂ G be a subgraph, then the distance between H and H ′ is given by d ( H, H ′ ) = min { d ( v, H ′ ) | v ∈ H } . In particular, d ( H, H ′ ) = 0 if and only if H ∩ H ′ = ∅ .(iv) Γ G ( H ) denotes the subset of vertices Γ G ( H ) = { v ∈ G | d ( v, H ) = 1 } . (v) In the case k > , S G,k ( H ) denotes the induced subgraph given by restrictingto the vertex set V ( S G,k ( H )) = { v ∈ G | d ( v, H ) ≥ k } . (vi) S G, denotes the subgraph given by the vertex set V ( S G, ( H )) = { v ∈ G | d ( v, H ) > or deg( v ) ≥ } . and the edge set E ( S G, ( H )) = { ( v, w ) ∈ E ( G ) | v, w ∈ V ( S G, ( H )) }\ { ( v, w ) ∈ E ( G ) | v, w ∈ H } . We clarify the previous notation in the following example.
Example 3.7. (i) Let G be the graph of Example 3.1 and H = C ∪ C be thesubgraph given by the two cycles of length . Then, we have that Γ G ( H ) is the set containing the vertex in the middle of the bridge joining the twocycles, that S G, ( H ) is a graph of the form x z y z z and that the graph z z EGULARITY OF BICYCLIC GRAPHS AND THEIR POWERS 15 represents S G, ( H ) .(ii) Let G be the graph given by x x x x x x x x x and H be the triangle induced by the vertices { x , x , x } . Then, we havethat Γ G ( H ) = { x , x , x } , that S G, ( H ) is a graph of the form x x x x x x x x x and that the graph x x x represents S G, ( H ) . We have already computed reg I ( G ) in the case n, m ≡ , Case I.
In this subsection we shall focus on the case n ≡ , m ≡ Notation 3.8.
Let G be a bicyclic graph with dumbbell C n · P l · C m such that n ≡ , and m ≡ . We shall denote by F , . . . , F c the connectedcomponents of S G, ( C m ) , and in this case each F i is either a tree or a unicyclicgraph with cycle C n (and n ≡ , ). Then, the graph S G, ( C m ) can begiven as the union of the components H , . . . , H c , where each one is defined as H i = F i \ { v ∈ G | d ( v, C m ) ≤ } . We note that each H i can be a non-connected graph or even the empty graph. Remark 3.9.
The following statements hold.(i) The graph G \ Γ G ( C m ) has a decomposition of the form G \ Γ G ( C m ) = C m [ c [ i =1 H i ! , and in particular ν ( G \ Γ G ( C m )) = ν ( C m ) + c X i =1 ν ( H i ) because d ( C m , H i ) ≥ for all ≤ i ≤ c and d ( H i , H j ) ≥ for all ≤ i Let G be the graph x x x z y z z y y y y y y and C be the cycle given by { y , y , y , y , y } . We have that Γ G ( C ) = { z , y } .The graph S G, ( C ) is given by x x x z y z z y y y with connected components F = { y , z , z , z , x , x , x } and F = { y , y , y } .The graph S G, ( C ) is given by x x x z z y and following our notations we have H = { x , x , x , z , z } and H = { y } . Lemma 3.11. Adopt Notation 3.8. If ν ( H i ) = ν ( F i ) for all ≤ i ≤ c , then ν ( G \ Γ G ( C m )) = ν ( G ) .Proof. Follows identically to [1, Lemma 3.5]. (cid:3) Proposition 3.12. Adopt Notation 3.8. If ν ( G \ Γ G ( C m )) < ν ( G ) then reg I ( G ) = ν ( G ) + 1 . EGULARITY OF BICYCLIC GRAPHS AND THEIR POWERS 17 Proof. Once more, we shall only prove that reg I ( G ) ≤ ν ( G ) + 1. Assume that ν ( G \ Γ G ( C m )) < ν ( G ), then the contrapositive of Lemma 3.11 implies that thereexists some i with ν ( H i ) < ν ( F i ).Fix i such that ν ( H i ) < ν ( F i ). From Remark 3.9( ii ), let x be the vertex in F i ∩ C m . Let us use the notations G ′ = G \ x and G ′′ = G \ N [ x ]. Again, we havethe inequality reg I ( G ) ≤ max { reg I ( G ′ ) , reg I ( G ′′ ) + 1 } . Note that both G ′ and G ′′ can be either unicyclic graphs with cycle C n (and n ≡ , I ( G ′ ) = ν ( G ′ ) + 1 and reg I ( G ′′ ) = ν ( G ′′ ) + 1 . In the case of G ′ , we have that reg I ( G ′ ) = ν ( G ′ ) + 1 ≤ ν ( G ) + 1. Let H bethe induced subgraph of G obtained by deleting the vertices of F i ∪ N G [ x ]. Thenwe have G ′′ = H ∪ H i . Let M and M be maximal induced matchings in H and H i , respectively, then ν ( G ′′ ) = |M | + |M | because d ( H, H i ) ≥ 2. By thecondition ν ( F i ) > ν ( H i ) then there exists a maximal induced matching M in F i ,such that |M | > |M | . From the fact that H ∪ F i is an induced subgraph in G and H ∩ F i = ∅ , then we get ν ( G ) ≥ ν ( H ∪ F i ) = |M | + |M | > |M | + |M | = ν ( G ′′ ) . Hence reg I ( G ′′ ) = ν ( G ′′ ) + 1 ≤ ν ( G ), and so we get the statement of the propo-sition. (cid:3) Theorem 3.13. Let G be a bicyclic graph with dumbbell C n · P l · C m such that n ≡ , and m ≡ . Then the following statements hold.(i) ν ( G ) + 1 ≤ reg I ( G ) ≤ ν ( G ) + 2 ;(ii) reg I ( G ) = ν ( G ) + 2 if and only if ν ( G ) = ν ( G \ Γ G ( C m )) .Proof. In Proposition 3.4 we proved ( i ). In order to prove ( ii ), we only need toshow that ν ( G \ Γ G ( C m )) = ν ( G ) implies reg I ( G ) ≥ ν ( G ) + 2, because the inverseimplication follows from Proposition 3.12.From Remark 3.9( i ), G \ Γ G ( C m ) = C m ∪ ( ∪ ci =1 H i ) where each H i is either aforest or a unicyclic graph with cycle C n (and n ≡ , I ( G \ Γ G ( C m )) = reg I ( C m ) + reg I ( ∪ ci =1 H i ) − 1= ( ν ( C m ) + 2) + ( ν ( ∪ ci =1 H i ) + 1) − ν ( G \ Γ G ( C m )) + 2= ν ( G ) + 2 . Finally, since G \ Γ G ( C m ) is an induced subgraph of G then we have reg I ( G ) ≥ ν ( G ) + 2. (cid:3) Case II. The object of study of this subsection is the case where n, m ≡ l ≥ I ( G ) = ν ( G ) + 3. More specifically, we shall givenecessary and sufficient conditions for the equality reg I ( G ) = ν ( G ) + 3. Notation 3.14. Let G be a bicyclic graph with dumbbell C n · P l · C m such that n, m ≡ and l ≥ . As in Notation 3.8, let F , . . . , F c be the componentsof the graph S G, ( C n ) . We order the F i ’s in such a way that F is a unicyclic graph with cycle C m , and for all i > we have that F i is a tree. The graph S G, ( C n ) can be decomposed in components H , . . . , H c where H i = F i \ { v ∈ G | d ( v, C n ) ≤ } . Remark 3.15. From the previous notation get the following simple remarks.(i) The graph G \ Γ G ( C n ) has a decomposition of the form G \ Γ G ( C n ) = C n [ c [ i =1 H i ! , and in particular ν ( G \ Γ G ( C n )) = ν ( C n ) + c X i =1 ν ( H i ) because d ( C n , H i ) ≥ for all ≤ i ≤ c and d ( H i , H j ) ≥ for all ≤ i Adopt Notation 3.14. If ν ( H i ) = ν ( F i ) for all ≤ i ≤ c and ν ( H ) = ν ( H \ Γ H ( C m )) , then ν ( G \ Γ G ( C n ∪ C m )) = ν ( G ) . Proof. Since G \ Γ G ( C n ∪ C m ) is an induced subgraph of G , then we have ν ( G \ Γ G ( C n ∪ C m )) ≤ ν ( G ). From Remark 3.15( ii ) we get ν ( G \ Γ G ( C n ∪ C m )) = ν ( C n ) + c X i =2 ν ( H i ) + ν ( H \ Γ H ( C m ))= ν ( C n ) + c X i =2 ν ( H i ) + ν ( H )= ν ( C n ) + c X i =1 ν ( F i ) ≥ ν ( G ) , and so ν ( G \ Γ G ( C n ∪ C m )) = ν ( G ). (cid:3) EGULARITY OF BICYCLIC GRAPHS AND THEIR POWERS 19 Proposition 3.17. Adopt Notation 3.14. If ν ( G \ Γ G ( C n ∪ C m )) < ν ( G ) , then reg I ( G ) ≤ ν ( G ) + 2 . Proof. It follows from the contrapositive of Lemma 3.16, that there exists some i with ν ( H i ) < ν ( F i ) or we have ν ( H \ Γ H ( C m )) < ν ( H ). Then we divide theproof into two cases.Case 1. In this case we assume that for some 1 ≤ i ≤ c we have ν ( H i ) < ν ( F i ).This case follows similarly to Proposition 3.12. Let x be the vertex in F i ∩ C n ,let us use the notations G ′ = G \ x and G ′′ = G \ N [ x ]. Once more, we have theinequality reg I ( G ) ≤ max { reg I ( G ′ ) , reg I ( G ′′ ) + 1 } . Note that both G ′ and G ′′ are unicyclic graphs, and so we have reg I ( G ′ ) ≤ ν ( G ′ )+2 and reg I ( G ′′ ) ≤ ν ( G ′′ ) + 2 (see Theorem 1.16). Since we have ν ( G ′ ) ≤ ν ( G ) and ν ( G ′′ ) + 1 ≤ ν ( G ) (see the proof of Proposition 3.12), then the inequality followsin this case.Case 2. Now we suppose that ν ( H \ Γ H ( C m )) < ν ( H ). Let x be the vertexin F ∩ C n , let us use the notations G ′ = G \ x and G ′′ = G \ N [ x ]. We use theinequality reg I ( G ) ≤ max { reg I ( G ′ ) , reg I ( G ′′ ) + 1 } . The graphs G ′ and G ′′ are unicyclic. For the graph G ′ we have reg I ( G ′ ) ≤ ν ( G ′ )+2 ≤ ν ( G ) + 2. The graph G ′′ can be given as the disjoint union of H and anothergraph H defined by H = G \ ( F ∪ N [ x ]), that is G ′′ = H ∪ H and H ∩ H = ∅ . Since H is a forest, then using [1, Theorem 1.1] we obtain that reg I ( G ′′ ) ≤ ν ( G ′′ ) + 1.So we get the inequality reg I ( G ′′ ) + 1 ≤ ν ( G ′′ ) + 2 ≤ ν ( G ) + 2, because G ′′ is aninduced subgraph of G . (cid:3) Now we are ready to completely describe the case where reg I ( G ) = ν ( G ) + 3. Theorem 3.18. Let G be a bicyclic graph with dumbbell C n · P l · C m . Then reg I ( G ) = ν ( G ) + 3 if and only if the following conditions are satisfied:(i) n ≡ ;(ii) m ≡ ;(iii) l ≥ ;(iv) ν ( G \ Γ G ( C n ∪ C m )) = ν ( G ) .Proof. In Proposition 3.4 we proved that the conditions ( i ), ( ii ) and ( iii ) are nec-essary, and from Proposition 3.17 we have that the condition ( iv ) is also necessary.Hence, we only need to prove that reg I ( G ) = ν ( G ) + 3 under these conditions.Let W = G \ Γ G ( C n ∪ C m ). From Remark 3.15, and using [1, Theorem 1.1] andTheorem 1.13, we can computereg (cid:0) I ( W ) (cid:1) = reg (cid:0) I ( C n ) (cid:1) + reg (cid:0) I (cid:0) ∪ ci =2 H i (cid:1)(cid:1) + reg (cid:0) I (cid:0) H \ Γ H ( C m ) (cid:1)(cid:1) − 2= ( ν ( C n ) + 2) + ( ν ( ∪ ci =2 H i ) + 1) + ( ν ( H \ Γ H ( C m )) + 2) − ν ( W ) + 3= ν ( G ) + 3 . Since W is an induced subgraph of G then we getreg I ( G ) ≥ reg I ( W )) = ν ( G ) + 3 , and so from Theorem 1.16 the equality it is obtained. (cid:3) Case III. In this subsection we assume that G is a bicyclic graph with dumbbell C n · P l · C m such that n, m ≡ l ≥ 3. Now that we have characterized whenreg I ( G ) = ν ( G ) + 3, then we want to distinguish between reg I ( G ) = ν ( G ) + 1and reg I ( G ) = ν ( G ) + 2. Lemma 3.19. Adopt Notation 3.14. If ν ( G ) − ν ( G \ Γ G ( C n ∪ C m )) = 1 then reg I ( G ) = ν ( G ) + 2 . Proof. From Theorem 3.18 we have that reg ( I ( G )) ≤ ν ( G ) + 2. Using the samemethod as in Theorem 3.18, we can obtain a lower boundreg I ( G ) ≥ reg I ( G \ Γ G ( C n ∪ C m )) = ν ( G \ Γ G ( C n ∪ C m )) + 3 = ν ( G ) + 2 , and so the equality follows. (cid:3) Lemma 3.20. Adopt Notation 3.14. If ν ( G ) = ν ( G \ Γ G ( C n )) then reg I ( G ) ≥ ν ( G ) + 2 . Symmetrically, the same argument holds for C m .Proof. The proof follows similarly to Theorem 3.13. From Remark 3.15( i ), [1,Theorem 1.1] and Theorem 1.13 we getreg I ( G \ Γ G ( C n )) = reg I ( C n ) + reg I ( ∪ ci =1 H i ) − 1= ( ν ( C n ) + 2) + ( ν ( ∪ ci =1 H i ) + 1) − ν ( G \ Γ G ( C n )) + 2= ν ( G ) + 2 . So the inequality follows from the fact that G \ Γ G ( C n ) is an induced subgraph of G . (cid:3) The following very simple logical argument will be used several times in thenext theorem. Observation 3.21. Let P , P , P be boolean values, (i.e. true or false). Assumethat P is true if and only if P and P are true, that is P ⇐⇒ ( P ∧ P ) . Suppose that if P is true then P is false, that is P = ⇒ ¬ P . Then, P is false. Notation 3.22. Let X be a mathematical expression. Then, P [ X ] represents aboolean value, which is true if X is satisfied and false otherwise. Taking into account the induced matching numbers ν ( G ), ν ( G \ Γ G ( C n ∪ C m )), ν ( G \ Γ G ( C n )) and ν ( G \ Γ G ( C m )), we can give necessary and sufficient conditionsfor the equality reg I ( G ) = ν ( G ) + 1. Theorem 3.23. Let G be a bicyclic graph with dumbbell C n · P l · C m such that n, m ≡ and l ≥ . Then reg I ( G ) = ν ( G ) + 1 if and only if the followingconditions are satisfied:(i) ν ( G ) − ν ( G \ Γ G ( C n ∪ C m )) > ;(ii) ν ( G ) > ν ( G \ Γ G ( C n )) ; EGULARITY OF BICYCLIC GRAPHS AND THEIR POWERS 21 (iii) ν ( G ) > ν ( G \ Γ G ( C m )) .Proof. From Lemma 3.19 and Lemma 3.20, we have that the conditions ( i ), ( ii )and ( iii ) are necessary. Hence, it is enough to prove reg I ( G ) ≤ ν ( G ) + 1 underthese conditions.Again, for any x ∈ G we denote G ′ = G \ x and G ′′ = G \ N [ x ]. We have theupper bound reg I ( G ) ≤ max { reg I ( G ′ ) , reg I ( G ′′ ) + 1 } . We shall prove that under the conditions ( i ), ( ii ) and ( iii ) there exists a vertex x ∈ C n such that reg I ( G ′ ) ≤ ν ( G ) + 1 and reg I ( G ′′ ) + 1 ≤ ν ( G ) + 1. We dividethe proof into three steps.Step 1. In this step we prove that for any x ∈ C n we have reg I ( G ′ ) ≤ ν ( G ) + 1.First we note the following two statements: • From Theorem 1.16 we have that reg I ( G ′ ) ≤ ν ( G ′ ) + 2. Hence, ν ( G ′ ) <ν ( G ) implies that reg I ( G ′ ) ≤ ν ( G ′ ) + 2 ≤ ν ( G ) + 1. • From [1, Theorem 1.1] we obtain that reg I ( G ′ ) = ν ( G ′ ) + 2 if and only if ν ( G ′ ) = ν ( G ′ \ Γ G ′ ( C m )).Thus, it follows thatreg I ( G ′ ) = ν ( G ) + 2 ⇐⇒ (cid:16) ν ( G ) = ν ( G ′ ) and ν ( G ′ ) = ν ( G ′ \ Γ G ′ ( C m )) (cid:17) . In Observation 3.21, let P = P (cid:2) reg I ( G ′ ) = ν ( G ) + 2 (cid:3) , P = P (cid:2) ν ( G ) = ν ( G ′ ) (cid:3) and P = (cid:2) ν ( G ′ ) = ν ( G ′ \ Γ G ′ ( C m )) (cid:3) . From the logical argument ofObservation 3.21, if we prove that ν ( G ′ ) = ν ( G ) implies ν ( G ′ ) > ν ( G ′ \ Γ G ′ ( C m ))then we will get the desired inequality reg I ( G ′ ) ≤ ν ( G ) + 1. Assume that ν ( G ) = ν ( G ′ ). From the hypothesis ν ( G ) > ν ( G \ Γ G ( C m )) and the fact that G ′ \ Γ G ′ ( C m ) is an induced subgraph of G \ Γ G ( C m ), then we get ν ( G ′ ) = ν ( G ) > ν ( G \ Γ G ( C m )) ≥ ν ( G ′ \ Γ G ′ ( C m )) . Therefore, we have reg I ( G ′ ) ≤ ν ( G ) + 1.Step 2. Since ν ( G ) > ν ( G \ Γ G ( C n )), it follows from Remark 3.15( iv ) that thereexists some 1 ≤ i ≤ c such that ν ( F i ) > ν ( H i ). Following Notation 3.14, we havethat F is a unicyclic graph containing the cycle C m and that F i is a tree for all i > 1. In this step, fix i > F i is a tree and ν ( F i ) > ν ( H i ).Let x be the vertex in F i ∩ C n and H be the induced subgraph H = G \ ( F i ∪ N G [ x ]). Note that G ′′ = H ∪ H i , d ( H, H i ) ≥ d ( H, F i ) ≥ 2. Then ν ( G ′′ ) = ν ( H ) + ν ( H i ) < ν ( H ) + ν ( F i ) ≤ ν ( G )follows from the condition ν ( H i ) < ν ( F i ). So we have that ν ( G ′′ ) < ν ( G ).Let K be the induced subgraph defined by K = ( G \ Γ G ( C m )) \ ( F i ∪ N [ x ]).Since i > F i ∩ F = ∅ , and so we get the following statements: • G ′′ \ Γ G ′′ ( C m ) = K ∪ H i . • K ∪ F i is an induced subgraph of G \ Γ G ( C m ). • We have the following inequalities ν ( G ′′ \ Γ G ′′ ( C m )) = ν ( K ) + ν ( H i ) < ν ( K ) + ν ( F i ) ≤ ν ( G \ Γ G ( C m )) . Again, as in Step 1, [1, Theorem 1.1] and Theorem 1.16 yield the followingequivalencereg I ( G ′′ )+1 = ν ( G )+2 ⇐⇒ (cid:16) ν ( G ) = ν ( G ′′ )+1 and ν ( G ′′ ) = ν ( G ′′ \ Γ G ′′ ( C m )) (cid:17) . In Observation 3.21, let P = P (cid:2) reg I ( G ′′ ) + 1 = ν ( G ) + 2 (cid:3) , P = P (cid:2) ν ( G ) = ν ( G ′′ + 1) (cid:3) and P = (cid:2) ν ( G ′′ ) = ν ( G ′′ \ Γ G ′ ( C m )) (cid:3) . So it is enough to prove that ν ( G ) = ν ( G ′′ ) + 1 implies ν ( G ′′ ) > ν ( G ′′ \ Γ G ′′ ( C m )). Assuming ν ( G ) = ν ( G ′′ ) + 1then we get ν ( G ′′ ) = ν ( G ) − > ν ( G \ Γ G ( C m )) − ≥ ν ( G ′′ \ Γ G ′′ ( C m )) . Therefore, in this case we have reg I ( G ′′ ) + 1 ≤ ν ( G ) + 1.Step 3. In this last step we assume that ν ( F ) > ν ( H ) and that ν ( F i ) = ν ( H i )for all i > 1. Let x be the vertex in F ∩ C n , then as in Step 2 we have thestatements: • ν ( G ′′ ) < ν ( G ). • reg I ( G ′′ ) + 1 = ν ( G ) + 2 ⇐⇒ (cid:16) ν ( G ) = ν ( G ′′ ) + 1 and ν ( G ′′ ) = ν ( G ′′ \ Γ G ′′ ( C m )) (cid:17) . Once more, if we prove that ν ( G ) = ν ( G ′′ ) + 1 implies ν ( G ′′ ) > ν ( G ′′ \ Γ G ′′ ( C m ))then we obtain that reg I ( G ′′ ) + 1 ≤ ν ( G ) + 1.We denote by L the induced subgraph of G ′′ \ Γ G ′′ ( C m ) given by disconnectingall the trees F i with i > 1, that is L = ( G ′′ \ Γ G ′′ ( C m )) \ Γ G ( C n ) . From the conditions ν ( F i ) = ν ( H i ) for all i > 1, then we get ν ( L ) = ν ( G ′′ \ Γ G ′′ ( C m )) (see the proofs of Lemma 3.11 or Lemma 3.16). We also have that L is an induced subgraph of G \ Γ G ( C n ∪ C m ) because we have the equality L = ( G \ Γ G ( C n ∪ C m )) \ N [ x ] . Finally, from the hypothesis ν ( G ) − ν ( G \ Γ G ( C n ∪ C m )) > ν ( G ′′ ) = ν ( G ) − > ν ( G \ Γ G ( C n ∪ C m )) ≥ ν ( L ) = ν ( G ′′ \ Γ G ′′ ( C m )) . Therefore, in this case we also have reg I ( G ′′ ) + 1 ≤ ν ( G ) + 1. (cid:3) Case IV. In this short subsection we deal with the remaining case, we assume that G is abicyclic graph with dumbbell C n · P l · C m such that n, m ≡ l ≤ l ≤ 2, the two cycles are too close to each other, and it is difficultto make a direct analysis (with our methods). Fortunately, using the completecharacterization of the case l ≥ 3, the problem can be solved with the Lozintransformation. Suppose that x is a vertex on the bridge P l (at most two), thenwe apply the Lozin transformation of G with respect to x , and obtain a bicyclicgraph L x ( G ) with dumbbell of the type C n · P k · C m where k ≥ 4. From [24, Lemma1] and [7, Theorem 1.1] we get the equality(4) reg ( I ( L x ( G ))) − ν ( L x ( G )) = reg ( I ( G )) − ν ( G ) . Therefore we obtain a characterization in the following corollary. Corollary 3.24. Let G be a bicyclic graph with dumbbell C n · P l · C m such that n, m ≡ and l ≤ . Let x be a point on the bridge P l and let L x ( G ) bethe Lozin transformation of G with respect to x . Then we have that ν ( G ) + 1 ≤ reg I ( G ) ≤ ν ( G ) + 2 , and that reg I ( G ) = ν ( G ) + 1 if and only if the followingconditions are satisfied:(i) ν ( L x ( G )) − ν ( L x ( G ) \ Γ L x ( G ) ( C n ∪ C m )) > ;(ii) ν ( L x ( G )) > ν ( L x ( G ) \ Γ L x ( G ) ( C n )) ; EGULARITY OF BICYCLIC GRAPHS AND THEIR POWERS 23 (iii) ν ( L x ( G )) > ν ( L x ( G ) \ Γ L x ( G ) ( C m )) .Proof. It follows from Proposition 3.4, (4), and Theorem 3.23. (cid:3) Examples. In this last subsection we shall give examples for each one of the statements inthe characterization of Theorem 3.2. Example 3.25. Statement (I) of Theorem 3.2. Let G be the graph below. x x x x z y y y z z Then we have reg I ( G ) = 4 and ν ( G ) = 3 . Example 3.26. Statement (II) of Theorem 3.2. Let G be the graph below. x x x y y y y y y y y Then we have reg I ( G ) = 5 and ν ( G ) = 3 .On the other hand, let G be the graph below. x x x y y y y y y y y z Then we have reg I ( G ) = 5 and ν ( G ) = 4 . Example 3.27. Statement (III) of Theorem 3.2. In Example 3.1 we saw a graph G where reg I ( G ) = 6 and ν ( G ) = 3 .Let G be the graph below. x x x x x z y y y y y z Then we have reg I ( G ) = 5 and ν ( G ) = 3 .But if we move the outer edge to the left, then we get a different result. Let G be the graph below. x x x x x z y y y y y z Then we have reg I ( G ) = 5 and ν ( G ) = 4 . Example 3.28. Statement (IV) of Theorem 3.2. Let G be the graph below. x x x x x y y y y Then we have reg I ( G ) = 4 and ν ( G ) = 2 .By adding an edge, let G be the graph below. x x x x x y y y y z Then we have reg I ( G ) = 4 and ν ( G ) = 3 . Castelnuovo-Mumford regularity of powers In this section, we study the regularity of the powers of I ( C n · P l · C m ) when l ≤ I ( C n · P l · C m ) q ,such that both coincide and are equal to 2 q + reg I ( C n · P l · C m ). To obtain the EGULARITY OF BICYCLIC GRAPHS AND THEIR POWERS 25 upper bound, we follow the argument of Banerjee from [4, Theorem 5.2]. To cal-culate the lower bound, we proceed by looking at “nice” induced subgraphs of C n · P l · C m .As a side result, we answer an interesting question on the behavior of the con-stant term of the asymptotically linear regularity function. Let I be an arbitraryideal generated in degree d and let b q := reg ( I q ) − dq for q ≥ 1. An interestingquestion is to study of the sequence { b i } i ≥ . In [11] Eisenbud and Harris provedthat if dim( R/I ) = 0, then { b i } i ≥ is a weakly decreasing sequence of non-negativeintegers. In [5] Banerjee, Beyarslan and H`a conjectured that for any edge ideal, { b i } i ≥ is a weakly decreasing sequence (see [5, Conjecture 7.11]). For the edgeideal of any dumbbell graph with l ≤ 2, we prove b i = b for all i ≥ 1. However,we expect b i ≤ b for all i ≥ Remark 4.1. From Theorem 2.4 and Theorem 2.6, for any l ≤ we have that reg I ( C n · P l · C m ) ≥ ⌊ n + m + l + 13 ⌋ . The previous inequality is not satisfied when l ≥ 3, because reg I ( C · P · C ) =3 and ⌊ ⌋ = 4.As recalled earlier, we use the notation of even-connection from Banerjee [4,Theorem 5.2]. The following lemma is crucial in our treatment of the even-connected vertices, and its proof is similar to [4, Lemma 6.13]. Lemma 4.2. Let G be a graph. As in Remark 1.10, let G ′ be the graph associatedto ( I ( G ) q +1 : e · · · e q ) pol . Suppose u = p , p , . . . , p s +1 = v is a path that even-connects u and v with respect to the q -fold e · · · e q . Then we have s +1 [ i =0 N G ′ [ p i ] ⊂ N G ′ [ u ] ∪ N G ′ [ v ] . Proof. Let U be the set of vertices U = { p , p , . . . , p s +1 } . For each 1 ≤ k ≤ s we have that p k − p k = e j k for some 1 ≤ j k ≤ q , i.e. u and v are even connectedwith respect to the s -fold e j e j · · · e j s .Let w be a vertex even-connected to some vertex z ∈ U with respect to the q -fold e · · · e q . Then, there exists a path z = r , r , . . . , r t +1 = w that even-connects z and w with respect to the q -fold e · · · e q . Let i be the largest integersuch that r i ∈ U . From the fact that r = z ∈ U , we have that the integer i iswell defined and i ≥ 0. Let k be an integer such that p k = r i .The proof is now divided into four different cases depending on i mod 2 and k mod 2. When i and k are both odd integers, we have that r i r i +1 is equal to someedge of { e , e , . . . , e q } and that p k − p k is not equal to any edge of { e j , e j , . . . e j s } .By the definition of i we have { r i +1 , r i +2 , . . . , r t +1 } ∩ U = ∅ . So, in this case, it follows that u = p , . . . , p k − , p k = r i , r i +1 , . . . , r t +1 = w is a path that even-connects u and w with respect to the q -fold e · · · e q .The other three cases follow in a similar way. (cid:3) Remark 4.3. Let G = C n · P l · C m . If ( I ( G ) q +1 : e · · · e q ) is not a square-freemonomial ideal and G ′ is the associated graph, then there exist a vertex x i which is even-connected to itself. Therefore G ′ has a leaf. By Lemma 4.2 one can see N G ′ [ x i ] contains one of the two cycles. In particular, if we denote the leaf by e ,then G ′ e is an induced subgraph of a unicyclic graph. Theorem 4.4. Let G = C n · P l · C m and I = I ( G ) be its edge ideal, then reg ( I q +1 : e · · · e q ) ≤ reg I for any ≤ q and any edges e , . . . , e q ∈ E ( G ) .Proof. We split the proof into two cases.Case 1. First, suppose ( I q +1 : e · · · e q ) is a square-free monomial ideal. In thiscase ( I q +1 : e · · · e q ) = I ( G ′ ) where G ′ is a graph with V ( G ) = V ( G ′ ) and E ( G ) ⊆ E ( G ′ ). Let E ( G ′ ) = E ( G ) ∪ { a , . . . , a r } . By Theorem 1.7, we havereg I ( G ′ ) ≤ max { reg I ( G ′ \ a ) , reg I ( G ′ a ) + 1 } From Lemma 4.2, G ′ a is obtained from G ′ by removing one of the cycles or deletingat least 6 vertices.Suppose G ′ a is obtained by removing one of the cycles. Without loss of gen-erality assume that C n is deleted, then there exists a Hamiltonian path of length ≤ m when l = 2 and of length ≤ m − l = 1. From Theorem 1.20 andRemark 4.1, if C n has n ≥ I ( G ′ a ) ≤ reg I ( G ) − 1. Inthe case n = 3, there is a Hamiltonian path of length ≤ m − 3, and so Theorem 1.20and Remark 4.1 again imply reg I ( G ′ a ) ≤ reg I ( G ) − G ′ a is obtained by removing at least 6 vertices. Let H ′ be the graphgiven by deleting N G [ a ]. From the assumption of deleting at least 6 vertices wehave that | H ′ | ≤ | G | − ≤ n + m + l − 8. We note that we can add two vertices to H ′ and connect them in such a way that we obtain a Hamiltonian path. Let H bea graph obtained by adding two vertices and certain edges connecting these twonew vertices, such that H has a Hamiltonian path. Note that G ′ a is an inducedsubgraph of H . Since | H | ≤ n + m + l − 6, Theorem 1.20 yieldsreg I ( H ) ≤ ⌊ n + m + l − ⌋ + 1 = ⌊ n + m + l + 13 ⌋ − . Applying Remark 4.1, we getreg I ( G ′ a ) ≤ reg I ( H ) ≤ reg I ( G ) − . Therefore reg I ( G ′ ) ≤ max { reg I ( G ′ \ a ) , reg I ( G ) } . In the same way, for any subgraph H = G ′ \ { a , . . . , a i } , we have thatreg ( I ( H a i +1 )) ≤ reg ( I ( G )) − . So, we also obtainreg I ( G ′ \ a ) ≤ max { reg I ( G ′ \ { a , a } ) , reg I ( G ) } . By continuing this process, we get reg I ( G ′ ) ≤ reg I ( G ).Case 2. Suppose ( I q +1 : e · · · e q ) is not square-free and G ′ is the graph associ-ated to ( I q +1 : e · · · e q ) pol . Let { b , b , . . . , b s } be the subset of edges of E ( G ′ ) \ E ( G ) that are generated by square monomials , i.e. each b i is a whisker.From Theorem 1.7 we have the inequalityreg I ( G ′ ) ≤ max { reg I ( G ′ \ b ) , I ( G ′ b ) } . EGULARITY OF BICYCLIC GRAPHS AND THEIR POWERS 27 Remark 4.3 implies that one of the cycles is deleted from G ′ b , then there existsan edge e ∈ G such that d ( e, G ′ b ) ≥ 2. So, for such an edge e we get that thedisjoint union G ′ b ∪ e is an induced subgraph of G ′ \ b . Thus, Theorem 1.7 and[19, Lemma 3.2] yield thatreg ( I ( G ′ b )) + 1 = reg ( I ( G ′ b ∪ e )) ≤ reg ( I ( G ′ )) . Therefore, we obtain that reg I ( G ′ ) ≤ reg I ( G ′ \ b ).By applying the same argument, it follows thatreg I ( G ′ ) ≤ reg I ( G ′ \ b ) ≤ reg I ( G ′ \ { b , b } ) ≤ · · · ≤ reg I ( G ′ \ { b , . . . , b s } ) . Since the graph G ′ \ { b , . . . , b s } has no whiskers, then Step 1 implies thatreg I ( G ′ ) ≤ reg I ( G ′ \ { b , . . . , b s } ) ≤ reg I ( G ) . Therefore, the proof is completed. (cid:3) Remark 4.5. The previous theorem is a generalization of a work done by YanGu in [14] for the case l = 1 . Theorem 4.6. For the dumbbell graph C n · P l · C m with l ≤ , we have reg I ( C n · P l · C m ) q ≥ q + reg I ( C n · P l · C m ) − , for any q ≥ .Proof. Using the inequality reg I ( C n · P · C m ) q ≥ q + ν ( C n · P · C m ) − I ( C n · P l · C m ) = ν ( C n · P l · C m ) + 1 weget the expected inequality. We divide the proof in two halves, the cases l = 1and l = 2.Case 1. Let l = 1. We only need to focus on the case where n, m ≡ H be the induced subgraph of C n · P · C m mentioned in the proof of Theorem 2.8,i.e. H = ( C n · P · C m ) \ { x n } = P n − · C m . Using Theorem 2.4, Proposition 2.3and the modularity n, m ≡ ν ( H ) = ν ( C n · P · C m )and that ν ( H ) = ν ( H \ Γ H ( C m )) . From Theorem 2.8 and [1, Theorem 1.1] we getreg I ( C n · P · C m ) = ν ( C n · P · C m ) + 2 = ν ( H ) + 2 = reg I ( H ) . Since H is an induced subgraph of C n · P · C m , then from [1, Theorem 1.2] and[6, Corollay 4.3] we get the inequalityreg I ( C n · P · C m ) q ≥ reg I ( H ) q = 2 q + reg I ( H ) − q + reg I ( C n · P · C m ) − . Case 2. Let l = 2. We only need to focus on the cases where n ≡ , m ≡ H as in Lemma 2.13.The induced subgraph H = ( C n · P · C m ) \ { x } of C n · P · C m is given as theunion of a path of length n − C m , i.e., H = P n − ∪ C m .By Theorem 2.14, for the cases n ≡ , m ≡ I ( C n · P · C m ) = ν ( C n · P · C m ) + 2 = ⌊ n ⌋ + ⌊ m ⌋ + 2 , and from [1, Theorem 1.1] we havereg I ( H ) = ν ( H ) + 2 = ν ( P n − ) + ν ( C m ) + 2 = ⌊ n ⌋ + ⌊ m ⌋ + 2 . Hence, we get reg I ( C n · P · C m ) = reg I ( H ). Finally, using [1, Theorem 1.2] and[6, Corollary 4.3], we get the inequalityreg I ( C n · P · C m ) q ≥ reg I ( H ) q = 2 q + reg I ( H ) − q + reg I ( C n · P · C m ) − . Therefore, the proof is completed. (cid:3) Theorem 4.7. For the dumbbell graph C n · P l · C m with l ≤ , we have reg I ( C n · P l · C m ) q = 2 q + reg I ( C n · P l · C m ) − for all q ≥ .Proof. It follows by Theorem 4.4, Theorem 1.11 and Theorem 4.6. (cid:3) Remark 4.8. One may ask whether reg I ( C n · P l · C m ) q = 2 q + reg I ( C n · P l · C m ) − always holds for given n, m, l and q . Unfortunately, this is not the case. In fact,it can be checked that I ( C · P · C ) < I ( C · P · C ) − . Acknowledgments This project is originated from the summer school “Pragmatic 2017”. Theauthors would like to sincerely express their gratitude to the organizers AlfioRagusa, Elena Guardo, Francesco Russo, and Giuseppe Zappal`a, and to the lec-turers Brian Harbourne, Adam Van Tuyl, Enrico Carlini, and T`ai Huy H`a. Weare deeply grateful to the last lecturer for introducing this topic to us and for hismentoring. We are grateful to the referee for valuable comments and suggestionsthat improved this paper in many ways. 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