A best bound for λ_2(G) to guarantee κ(G) \geq 2
aa r X i v : . [ m a t h . C O ] J a n A best bound for λ ( G ) to guarantee κ ( G ) ≥ Wenqian Zhang, Jianfeng Wang ∗ School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, China
Abstract
Let G be a connected d -regular graph with a given order and the second largest eigen-value λ ( G ). Mohar and O (private communication) asked a challenging problem: what isthe best upper bound for λ ( G ) which guarantees that κ ( G ) ≥ t + 1, where 1 ≤ t ≤ d − κ ( G ) is the vertex-connectivity of G , which was also mentioned by Cioab˘a. As a startingpoint, we solve this problem in the case t = 1, and characterize all families of extremal graphs. AMS classification:
Keywords : Regular graphs; Second largest eigenvalue; Cut vertex; Connectivity.
In this paper, we focus on the eigenvalues d > λ ( G ) ≥ · · · ≥ λ n ( G ) of adjacency matrix A ( G )of d -regular graphs with order n . Set λ = λ ( G ) = max {| λ ( G ) | , | λ n ( G ) |} to be the secondeigenvalue of G . Proverbially, the second largest eigenvalue λ and the second eigenvalue λ ofgraphs have been paid much attention in several aspects. One of the best-known cases is that itcould describe the expander graphs which are useful in the design and analysis of communicationnetworks. From the point view of spectral graph theory, it was Alon and Milman [4] and Dodziuk[17] who independently gave a discrete analogue of Cheeger’s result on Riemannian manifold [9],which is stated as follows and involves the edge expansion ratio h ( G ) of G . d − λ ≤ h ( G ) ≤ p d ( d − λ ) , where h ( G ) = min S ⊆ V, | S |≤| V | / | E ( S, S ) || S | . (1)As we have known, large spectral gap d − λ implies high expansion, vice versa. Remarkably,Alon [2] showed that a regular bipartite graph is an expander iff d − λ is large enough (note λ = λ for bipartite graphs). Further, we precisely know how large the spectral gap can be in an d -regular graph G n,d of order n , due to Alon-Boppana bound lim n →∞ inf λ ( G n,d ) = 2 √ d − d -regulargraph G is Ramanujan if λ ( G ) ≤ √ d −
1. The significant progress in this field may be thatthere exist infinite families of regular bipartite Ramanujan graphs of every degree bigger than 2,owe to Marcus et al [27]. On the other side, it is also a rather difficult problem that determiningall the connected (regular) graphs with small λ ( ≤ √ d − λ ≤ reflexive graphs corresponding to sets of vectors norm 2 and at angles 90 ◦ or120 ◦ but without a structural characterization, thanks to Neumaier and Seidel [16]. Especially, ∗ Corresponding author.
Email addresses: [email protected](W.Q. Zhang), [email protected] (J.F.Wang).
Hyperbolic trees . For other miscellaneous resultsabout the graphs with small λ ≤ or √ − √ − , see Cvetkovi´c and Simi´c’s survey [10]and [34, 35, eg.] for (regular) graphs with λ ≤ κ ′ ( G ) of a d -regulargraph based on its second largest eigenvalue λ ( G ), there are few ones concerning the vertex-connectivity κ ( G ) built on λ ( G ), the earliest two of which are respectively Krivelevich andSudakov’s Theorem 4.1 in [23] and Fiedler’s Theorem 4.1 [18] (note, λ ( G ) = d − α ( G ) with α ( G ) being the algebraic connectivity). The others are obtained in past five years. Some ofthem is fit for the regular multigraph [1, 31]. Here we concentrate on the results for the regularsimple graphs, the first one of which is due to Cioab˘a and Gu [13]. Proposition 1.1. [13, Cioab˘a and Gu]
Let G be a connected d -regular simple graph, d ≥ , and λ ( G ) < ( d − √ d +122 if d is even d − √ d +82 if d is old . Then κ ( G ) ≥ . Proposition 1.2. [1, Abiad et al.]
Let k be an integer and G be a d -regular graph of order n with d ≥ k ≥ . Set f ( d, k ) = d + 1 if k = 2 and f ( d, k ) = d + 2 − k otherwise. If λ ( G ) < d − ( k − dn f ( d,k )( n − f ( d,k )) , then κ ( G ) ≥ k . Proposition 1.3. [25, Liu]
Let k be an integer and G be a d -regular graph of order n with d ≥ k ≥ . Set β = ⌈ ( d + 1 + p ( d + 1) − k − d ) ⌉ and ϕ ( d, k ) = ( d + 1)( n − d − if k = 2( d − k + 2)( n − d + k − if k ≥ and d ≤ k − β ( n − β ) if k ≥ and d > k − . If λ ( G ) < d − ( d − nd ϕ ( d,k ) , then κ ( G ) ≥ k . Hong et al. [20] found their bound for λ ( G ) improving the previous two ones. Proposition 1.4. [20, Hong et al.]
Let k be an integer and G be a d -regular graph of order n with d ≥ k ≥ . If λ ( G ) < d − ( k − nd ( n − k +1)( k − d − k +2)( n − d − , then κ ( G ) ≥ k . As we’ve already seen, the above bounds for λ ( G ) are not sharp. Then, the aim of thepresent paper is to find the best upper bounds on the second largest eigenvalues of regulargraphs guaranteeing a desired vertex-connectivity. In other words, we investigate the challengingquestion asked by Mohar and O (private communication) and alluded to briefly by Cioab˘a [11, 13]and raised formally by Abiad et al. [1]. Problem 1.
For a d -regular simple graph or multigraph G and for ≤ t ≤ d − , what is thebest upper bound for λ ( G ) which guarantees that κ ( G ) ≥ t + 1 or that κ ′ ( G ) ≥ t + 1 ? κ ′ ( G ), that Cioab˘a [11] proved the cases t = 1 , t ≥ t = 1, and characterize all families of extremal graphs.To describe our results, we introduce some notations and terminology. As usual, let C n , K n and M n be the cycle , the complete graph and the perfect matching with order n . The sequentialjoin G ∨ · · · ∨ G k of graphs G , . . . , G k is the graph formed by taking one copy of each graphand adding additional edges from each vertex of G i to all vertices of G i +1 , for 1 ≤ i ≤ k − S, T ⊆ V ( G ), let E ( S, T ) = { ( u, v ) | u ∈ S, v ∈ T, ( u, v ) ∈ E ( G ) } be the set of edges from S to T . The graph G − S is derived from G by deleting the vertices of S and the edges incidentwith the vertices in S . If S = { v } , we denote by G − v for short. For two integers d ≥ ≤ c ≤ d −
1, we define the following set G d,c and graph G d,c relating to our main results. • G d,c is the set of connected d -regular graphs G with a cut vertex (say u ) such that G − u has a component G with | E ( u, V ( G )) | = c . • Let d ≥ G d,c = K ∨ M d − ∨ K ∨ K ∨ M d − ∨ K if c = 1 or c = d − M d +2 − c ∨ C c ∨ K ∨ M d − c ∨ K c +1 if c ∈ (2 , d −
2] is odd; K d +1 − c ∨ M c ∨ K ∨ C d − c ∨ M c +2 if c ∈ [2 , d −
2) is even , where C c = C c ∪ · · · ∪ C c s is the union of disjoint cycles C c i and P si =1 c i = c . • Let d ≥ c ∈ [2 , d −
2] is even, and define G d,c = K d +1 − c ∨ M c ∨ K ∨ M d − c ∨ K c +1 . Remark that G d, = G d,d − and G d,c ∈ G d,c .We are now in the stage to present the main result of this paper. Theorem 1.5.
Let d ≥ and G be a connected d -regular graph. (i) Let d = 3 . If λ ( G ) ≤ λ ( G d, ) and G = G d, , then κ ( G ) ≥ . (ii) Let d ≥ be even. If λ ( G ) ≤ λ ( G d, ⌊ d ⌋ ) and G = G d, ⌊ d ⌋ = K ⌊ d +24 ⌋ +1 ∨ M ⌊ d ⌋ ∨ K ∨ M ⌊ d +24 ⌋ ∨ K ⌊ d ⌋ +1 , then κ ( G ) ≥ . (iii) Let d ≥ be odd and C c be defined as above. (a) For odd d − , if λ ( G ) ≤ λ ( G d, d − ) and G = G d, d − = M d +52 ∨ C d − ∨ K ∨ M d +12 ∨ K d +12 , then κ ( G ) ≥ . (b) For even d − , if λ ( G ) ≤ λ ( G d, d − ) and G = G d, d − = K d +32 ∨ M d − ∨ K ∨ C d +12 ∨ M d +32 , then κ ( G ) ≥ . Preparation
Let M n ( F ) be the set of all n by n matrices over a field F . A matrix M = [ m ij ] ∈ M n ( F ) is tridiagonal if m ij = 0, whenever | i − j | > Lemma 2.1. [6, 21]
Let M be a non-negative tridiagonal matrix as follows: A = a b c a b . . . . . . . . .. . . . . . b n − c n a n ,Assume each row sum of M equals d . If M has eigenvalues λ = d, λ , ..., λ n +1 indexed innon-increasing order, then the n × n matrix e A = d − b − c b c d − b − c b c . . . . . .. . . . . . b n − c n − d − b n − − c n has eigenvalues λ , λ , ..., λ n +1 . For the adjacency matrix A ( G ), an equitable partition of a graph G is a partition of thevertex set V ( G ) into parts V i such that each vertex in V i has the same number b i,j of neighborsin part V j for any j . Then the matrix B = ( b i,j ) is called the quotient matrix of G w.r.t. thegiven partition. Lemma 2.2. [7]
Let B be the quotient matrix of a graph G w.r.t. the partition { V , . . . , V m } .Then the eigenvalues of B interlace the eigenvalues of G . Moreover, if this partition is equitable,then each eigenvalue (with multiplicity) of B is an eigenvalue of G . Look back to the graph G d,c ∈ G d,c . Obviously, G d,c contains 2 d + 4 vertices for odd d ≥ d + 3 vertices for even d ≥
4. Moreover, G d,c = G d,d − c . Note that G d, = G d,d − is the graph X d defined in [11, (4)] and used in [11, Theorem 1.4], the proof of which impliesthe following conclusion. Lemma 2.3. [11]
Let d ≥ be an odd integer. Then λ ( G d, )(= λ ( G d,d − )) is the largest rootof f ( x ) = 0 with f ( x ) = x − ( d − x − (3 d − x − . Moreover, if G is a d -regular graphwith cut edges, then λ ( G ) ≥ λ ( G d, ) with equality if and only if G = G d, . Lemma 2.4.
Let G d,c be the graph defined as above. (i) Let d ≥ and ≤ c ≤ d − be even integers. Then λ ( G d,c ) is the largest root of equation f ( x ) = 0 , where f ( x ) = x − ( d − x − (4 d − x + (2 cd − c − d ) x + 3 c ( d − c ) . (ii) Let d ≥ be an odd integer and ≤ c ≤ d − . Then λ ( G d,c ) is the largest root of equation f ( x ) = 0 , where f ( x ) = x − ( d − x − (5 d − x + (2 cd − c − d ) x + 4 c ( d − c ) .Proof. For (i), G d,c = K d +1 − c ∨ M c ∨ K ∨ M d − c ∨ K c +1 has an equitable partition V ( G d,c ) = V ( K d +1 − c ) ∪ V ( M c ) ∪ V ( K ) ∪ V ( M d − c ) ∪ V ( K c +1 ). Then the corresponding quotient matrix is B = d − c c d + 1 − c c − c d − c
00 0 1 d − c − c + 10 0 0 d − c c , B are the eigenvalues of G d,c . ByLemma 2.1 we have λ ( B ) = λ ( f B ), where f B = − d + 1 − c d − c − d − c c c − c + 10 0 1 − . Thus, a direct calculation shows that λ ( f B ) is the largest root of equation f ( x ) = 0. Note λ ( f B ) > d − . Let W be the characteristic space of the five parts (of the equitable partition)of G d,c . Clearly, the dimension of W is five. Note that each eigenvector x associated to aneigenvalue of B can be extended to be an eigenvector x ′ of G d,c , and that the components of x ′ corresponding to each part of the equitable partition are the same. Hence, the five independenteigenvectors extended by those of B span W . Except eigenvalues of B , remark again that theeigenvectors of other eigenvalues of G d,c are orthogonal to those in W . So, other eigenvalues of G d,c are also the eigenvalues of graph K d +1 − c ∪ M c ∪ K ∪ M d − c S K c +1 , whose maximum degreeis not larger than d −
1. Hence, other eigenvalues of G d,c except the ones of B are less than d −
1. Thereby, λ ( G d,c ) = λ ( B ) = λ ( f B ), which is the largest root of equation f ( x ) = 0.For (ii), consider firstly c ∈ [2 , d −
2] to be an odd integer. Clearly, G d,a = M d +2 − c ∨ C c ∨ K ∨ M d − c ∨ K c +1 has an equitable partition V ( M d +2 − c ) ∪ V ( C c ) ∨ V ( K ) ∨ V ( M d − c ) ∨ V ( K c +1 ).Then, the related quotient matrix is B = d − c c d + 2 − c c − c d − c
00 0 1 d − c − c + 10 0 0 d − c c . From Lemma 2.2 the eigenvalues of B are also the eigenvalues of G d,c . By Lemma 2.1 again weget λ ( B ) = λ ( f B ), where f B = − d + 2 − c d − c − d − c c c − c + 10 0 1 − . (2)By a routine computing, λ ( f B ) is the largest root of equation f ( x ) = 0. Note that λ ( f B ) >d − . Similarly to (i), we can verify that λ ( G d,c ) = λ ( B ) = λ ( f B ) is the largest root ofequation f ( x ) = 0. If 2 ≤ c ≤ d − ≤ d − c ≤ d − G d,c = G d,d − c ,applying the above discussion to G d,d − c we obtain the conclusion. Lemma 2.5.
For d ≥ and ≤ c ≤ k − , let G k,a be the graph defined above. (i) If d ≥ is even, then λ ( G d, ) > λ ( G d, ) > · · · > λ ( G d, ⌊ d ⌋ ) . (ii) If d ≥ is odd, then λ ( G d, ) > λ ( G d, ) > · · · > λ ( G d, d − ) .Proof. Let d ≥ ≤ c ≤ ⌊ d ⌋ . By Lemma 2.4(i), λ ( G k,a ) is the largest root ofequation f ( x ) = 0. Since f ( x ) = x − ( d − x − (4 d − x + (2 cd − d − k ) x + 3 c ( d − c )= x − ( d − x − (4 d − x − dx + c ( d − c )(2 x + 3) , x > c ( d − c )(2 x + 3) is strictly increasing w.r.t. c . Thereby,the largest root of equation f ( x ) = 0 is strictly decreasing w.r.t. c , which implies λ ( G d, ) >λ ( G d, ) > · · · > λ ( G d, ⌊ d ⌋ ).We next show (ii). For d = 3, the result is trivial. Let d ≥ ≤ c ≤ k − . ByLemma 2.4(ii), λ ( G d,c ) is the largest root of equation f ( x ) = 0. Since f ( x ) = x − ( d − x − (5 d − x + (2 cd − c − d ) x + 4 c ( d − c )= x − ( d − x − (5 d − x − dx + 2 c ( d − c )( x + 2) , then for x > c ( d − c )( x + 2) is strictly increasing w.r.t. c . Therefore, the largestroot of equation f ( x ) = 0 is strictly decreasing w.r.t. c , and hence λ ( G d, ) > λ ( G d, ) > · · · >λ ( G d, d − ).For c = 1, from Lemma 2.3 it follows that λ ( G d, ) is the largest root of f ( x ) = 0. ByLemma 2.4(ii) again, λ ( G d, ) is the largest root of equation f ( x ) | c =2 = x − ( d − x − (5 d − x − (2 d + 8) x + 8 d −
16= ( x + 2)( x − ( d − x − (3 d − x −
2) + 2 x (2 d − − x ) + 8 d −
12= 0 . For d ≥ λ ( G k, ) ≤ x < d , we get f ( x ) ≥
0, and then x − ( d − x − (5 d − x − (2 d +8) x + 8 d − >
0. Hence, λ ( G d, ) > λ ( G d, ), and so λ ( G k, ) > λ ( G k, ) > · · · > λ ( G k, k − ).This completes the proof. Recall, for two integers d ≥ ≤ c ≤ d −
1, that the family G d,c is defined in Section 1. Lemma 3.1.
Let G ∈ G d,c . If d ≥ is odd and c = 1 or c = d − , then λ ( G ) ≥ λ ( G d,c ) withequality holds if and only if G = G d,c .Proof. In this case, G has cut edges. Then the result follows from Lemma 2.3. Lemma 3.2.
Let G ∈ G d,c . If d ≥ and c ∈ [2 , d − are even , then λ ( G ) ≥ λ ( G d,c ) , wherehe equality holds if and only if G = G d,c .Proof. Since G ∈ G d,c , then there exists a cut vertex u such that | E ( u, V ( G )) | = c , where G is a component of G − u . Let G be the union of other components of G − u . Then | E ( u, V ( G )) | = d − c . If | V ( G ) | ≤ d , then c = | E ( u, V ( G )) | ≥ | V ( G ) | ( d + 1 − | V ( G ) | ) ≥ d , acontradiction. Hence, | V ( G ) | ≥ d + 1, and so is | V ( G ) | similarly. Let V and V be the sets ofthe neighbors of u in G and in G respectively. Then | V | = c and | V | = d − c . Set G ′ = G − V and G ′ = G − V . Thus, | V ( G ′ ) | = p ≥ d + 1 − c and | V ( G ′ ) | = q ≥ d + 1 − ( d − c ) = c + 1.After putting | E ( V ( G ′ ) , V ) | = r and | E ( V ( G ′ ) , V ) | = t , we obtain a partition of G with V ( G ) = V ( G ′ ) ∪ V ∪ { u } ∪ V ∪ V ( G ′ ) whose quotient matrix is B = d − rp rp rc d − − rc c d − c
00 0 1 d − − td − c td − c tq d − tq . (3)6sing Lemmas 2.1 and 2.2 we get λ ( G ) ≥ λ ( B ) = λ ( f B ), where f B is d − rp − rc rc d − c − d − c c c − td − c d − td − c − tq . (4) Fact 1.
The largest root λ ( f B ) of matrix f B is strictly decreasing w.r.t. r and t . Proof of Fact 1.
We only prove the conclusion for r , as it is similar for t . By a computing, thecharacteristic polynomial of f B is equal to h ( x ) = | xI − f B | = ( x − d ) h ( x ) + rh ( x ), where h ( x ) = ( p + c ) h ( x ) + tc ( d − c ) − c ( x + c ( d − − c ) + 1)( x − d + td − c + tq ), and h ( x ) is the nextcharacteristic polynomial of a principle sub-matrix of f B d − c − d − c c c − td − c d − td − c − tq . Clearly, for x ≥ λ ( f B ) we get h ( x ) > h ( x ) ≥
0. Thus, for λ ( f B ) ≤ x < d we arrive at h ( x ) > x − d ) h ( x ) + r h ( x ) > h ( x ) ≥ r > r . Thereby, the largestroot of equation ( x − d ) h ( x ) + r h ( x ) = 0 is less than λ ( f B ). So, the largest root of f B isstrictly decreasing w.r.t. r . ✷ Employing Fact 1, we can make r and t as large as possible. By the well-know Perron-Frobenius Theorem, the largest eigenvalue of an irreducible matrix will strictly decrease if itspositive elements decrease. Then for given r and t we set p and q as small as possible in f B .Noting r ≤ c ( d −
1) and t ≤ ( d − c )( d − d +1 − c ≤ p ≤ d − c +1 ≤ q ≤ d − r = cp and t = ( d − c ) q (For example, if p ≥ d , we can firstly make r as large as possible, i.e., r = c ( d − p as small as possible, i.e., p = d − G [ V ] , G [ V ( G ′ )] , G [ V ] and G [ V ( G ′ )] are four regular graphs of degrees d − − p, d − c, d − − q and c , respectively (notethat for any two positive integers d < t , then there exists a d -regular graph on t vertices if d iseven). Apparently, f B is equal to the following matrix DD = d − c − p p d − c − d − c c c − q c − q . (5) Fact 2 . The largest root λ ( D ) of matrix D is a strictly increasing function w.r.t. p and q . Proof of Fact 2.
We only prove the conclusion for p , as it is similar for q . A straightforwardcalculation yields the characteristic polynomial of D is equal to g ( x ) = | xI − D | = ( x − d + c ) g ( x ) + pg ( x ), where g ( x ) = g ( x ) − ( x − c + 1)( x − c + q ), and g ( x ) is the characteristicpolynomial of next principle sub-matrix of D , d − c − d − c c c − q c − q . For x ≥ λ ( D ), clearly we get g ( x ) >
0. For any d + 1 − c ≤ p ′ < p , we now prove ( x − d + c ) g ( x ) + p ′ g ( x ) > x ≥ λ ( D ). Evidently, for any λ ( D ) ≤ x < d , if g ( x ) ≥
0, then( x − d + c ) g ( x ) + p ′ g ( x ) >
0. If g ( x ) < λ ( D ) ≤ x < d , then ( x − d + c ) g ( x ) + p ′ g ( x ) > ( x − d + c ) g ( x ) + pg ( x ) ≥
0. Hence, ( x − d + c ) g ( x ) + p ′ g ( x ) > x ≥ λ ( D ),7hich implies that the largest root of equation ( x − d + c ) g ( x ) + p ′ g ( x ) = 0 is less than λ ( D ).So, the largest root λ ( D ) of matrix D is strictly increasing w.r.t. p . ✷ In view of Fact 2, we can set p = d + 1 − c and q = c + 1 which together with r = cp and t = ( d − c ) q leads to f B = f B (defined in Lemma 2.4). Consequently, λ ( G ) ≥ λ ( G d,c ) withequality holds if and only if G = G d,c .For the completeness of next lemma, we give an imitate proof with the similar as Lemma3.2. Lemma 3.3.
Let G ∈ G d,c and d ≥ be an odd integer. (i) If c ∈ [2 , d − is odd, then λ ( G ) ≥ λ ( G d,c ) with equality if and only if G = M d +2 − c ∨ C c ∨ K ∨ M d − c ∨ K c +1 , where C c is the union of disjoint cycles on c vertices. (ii) If c ∈ [2 , d − is even, then λ ( G ) ≥ λ ( G d,c ) with equality if and only if G = K d +1 − c ∨ M c ∨ K ∨ C d − c ∨ M c +2 , where C d − c is the union of disjoint cycles on d − c vertices.Proof. We only need to show (i). Otherwise, if c ∈ [2 , d −
2] is even, then d − c is odd. Due to G d,c = G d,d − c , then the proof is similar to the case in which c is odd.Since G ∈ G d,c , then there exists a cut vertex u such that | E ( u, V ( G )) | = c , where G is acomponent of G − u . Let G be the union of other components of G − u . Then | E ( u, V ( G )) | = d − c . Similarly to Lemma 3.2, we have | V ( G ) | ≥ d + 1 and | V ( G ) | ≥ d + 1. Since d | V ( G ) | =2 | E ( G ) | + c and d | V ( G ) | = 2 | E ( G ) | + d − c , then | V ( G ) | is odd, and thus | V ( G ) | ≥ d + 2 and | V ( G ) | is even. Let V and V be the sets of the neighbors of u in G and in G respectively.Then | V | = c and | V | = d − c . Set G ′ = G − V and G ′ = G − V . Thus, | V ( G ′ ) | = p ≥ d +2 − c and | V ( G ′ ) | = q ≥ d + 1 − ( d − c ) = c + 1. Set | E ( V ( G ′ ) , V ) | = r and | E ( V ( G ′ ) , V ) | = t . Weobtain a partition of G with V ( G ) = V ( G ′ ) ∪ V ∪ { u } ∪ V ∪ V ( G ′ ) whose quotient matrix isjust B in (3). By Lemmas 2.1 and 2.2, we have λ ( G ) ≥ λ ( B ) = λ ( f B ), where f B is definedin (4). As shown in Lemma 3.2, by Fact 1 we get that the largest root λ ( f B ) of matrix f B isstrictly decreasing w.r.t. r and t .By the well-known Perron-Frobenius Theorem, the largest eigenvalue of an irreducible matrixwill strictly decrease if its positive elements decrease. Then we can make p and q as small aspossible, and set r and t as large as possible. Noting r ≤ c ( d −
1) and t ≤ ( d − c )( d − d + 2 − c ≤ p ≤ d − c + 1 ≤ q ≤ d − r = cp and t = ( d − c ) q . Then G [ V ( G )] , G [ V ( G ′ )] , G [ V ] , G [ V ( G ′ )] are four regular graphs of degree d − − p, d − c, d − − q and c , respectively (note that such graphs exist since p and q are even). Thus, f B is equal tomatrix D defined in (5).From Fact 2 it follows that the largest root λ ( D ) is strictly increasing w.r.t p and q . Thus,we can set p = d + 2 − c and q = c + 1, which implies f B = f B defined in (2). Consequently, λ ( G ) ≥ λ ( G d,c ), and equality holds if and only if G = M d +2 − c ∨ C c ∨ K ∨ M d − c ∨ K c +1 , where C c is the union of disjoint cycles on c vertices. Proof of Theorem 1.5 . Clearly, (i) follows from Lemma 2.3. For (ii) and (ii), assume that G has a cut vertex, say u . Then, there exists some component G of G − u such that | E ( u, V ( G )) | = c ≤ d − if d is odd, or such that c ≤ ⌊ d ⌋ if d is even. From Lemmas 3.1–3.3 and 2.5 we get λ ( G ) ≥ λ ( G d, d − ) with the equality iff G = G d, d − when d is odd, or λ ( G ) ≥ λ ( G d, ⌊ d ⌋ )with the equality iff G = G d, ⌊ d ⌋ when d is even, a contradiction.Consequently, κ ( G ) ≥ ✷ cknowledgments The authors are supported for this research by the National Natural Science Foundation ofChina (No. 11971274).
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