The relation between the independence number and rank of a signed graph
aa r X i v : . [ m a t h . C O ] J u l The relation between the independence number and rank of asigned graph
Shengjie He , Rong-Xia Hao ∗
1. Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, China
Abstract
A signed graph ( G, σ ) is a graph with a sign attached to each of its edges, where G isthe underlying graph of ( G, σ ) . Let c ( G ) , α ( G ) and r ( G, σ ) be the cyclomatic number, theindependence number and the rank of the adjacency matrix of ( G, σ ) , respectively. In thispaper, we study the relation among the independence number, the rank and the cyclomaticnumber of a signed graph ( G, σ ) with order n , and prove that n − c ( G ) ≤ r ( G, σ )+2 α ( G ) ≤ n . Furthermore, the signed graphs that reaching the lower bound are investigated. Keywords : Independence number; Cyclomatic number; Rank; Signed graph.
MSC : 05C50
All graphs considered in this paper are simple and finite. Let G = ( V, E ) be an undirected graphwith V = { v , v , · · · , v n } is the vertex set and E is the edge set. For a vertex u ∈ V ( G ) , the degree of u , denote by d G ( u ) , is the number of vertices which are adjacent to u . A vertex of G is called a pendant vertex if it is a vertex of degree one in G , whereas a vertex of G is calleda quasi-pendant vertex if it is adjacent to a pendant vertex in G unless it is a pendant vertex.Denote by P n , S n and C n a path, star and cycle on n vertices, respectively. The adjacency matrix A ( G ) of G is an n × n matrix whose ( i, j ) -entry equals to 1 if vertices v i and v j are adjacent and0 otherwise. We refer to [3] for undefined terminologies and notation.A signed graph ( G, σ ) consists of a simple graph G = ( V, E ) , referred to as its underly-ing graph, and a mapping σ : E → { + , −} , its edge labelling. To avoid confusion, we oftenwrite V ( G ) and E ( G ) for V ( G, σ ) and E ( G, σ ) , respectively. The adjacency matrix of ( G, σ ) is A ( G, σ ) = ( a σij ) with a σij = σ ( v i v j ) a ij , where ( a ij ) is the adjacent matrix of the underlying graph G . Let ( G, σ ) be a signed graph. An edge e ′ is said to be positive or negative if σ ( e ′ ) = + or σ ( e ′ ) = − , respectively. In the case of σ = + , which is an all-positive edge labelling, A ( G, +) is exactly the classical adjacency matrix of G . Thus a simple graph can always be viewed asa singed graph with all positive edges. Let C be a cycle of ( G, σ ) . The sign of C is definedby σ ( C ) = Q e ∈ C σ ( e ) . A cycle C is said to be positive or negative if σ ( C ) = + or σ ( C ) = − ,respectively. By definition, a cycle C is positive if and only if it has even number of negativeedges. The rank of a signed graph ( G, σ ) , written as r ( G, σ ) , is defined to be the rank of its ∗ Corresponding author. Emails: [email protected] (Shengjie He), [email protected] (Rong-Xia Hao) A ( G, σ ) . The nullity of ( G, σ ) is the multiplicity of the zero eigenvalues of A ( G, σ ) .A subset I of V ( G ) is called an independent set if any two vertices of I are independentin a graph G . The independence number of G , denoted by α ( G ) , is the number of vertices ina maximum independent set of G . Let c ( G ) be the cyclomatic number of a graph G , that is c ( G ) = | E ( G ) | − | V ( G ) | + ω ( G ) , where ω ( G ) is the number of connected components of G . The matching number of G , denoted by m ( G ) , is the cardinality of a maximum matching of G . Fora signed graph ( G, σ ) , the independence number, cyclomatic number and matching number of ( G, σ ) are defined to be the independence number, cyclomatic number and matching number ofits underlying graph, respectively.If G is a graph which any two cycles (if any) of G have no vertices in common. Denoted by C G the set of all cycles of G . Contracting each cycle of G into a vertex (called a cyclic vertex ),we obtain a forest denoted by T G . Denoted by O G the set of all cyclic vertex of G . Moreover,denoted by [ T G ] the subgraph of T G induced by all non-cyclic vertices. It is obviously that [ T G ] = T G − O G .The study on rank and nullity of graphs is a major and heated issue in graph theory. In[19], the relation between the independence number and the rank of a graph was investigatedby Wang and Wong. Gutman et al. [7] researched the nullity of line graphs of trees. Moharet al. characterized the properties of the H -rank of mixed graphs in [8]. Li et at. studied therelation among the rank, the matching number and the cyclomatic number of oriented graphsand mixed graphs in [10] and [13], respectively. Bevis et al. [2] obtained some results examiningseveral cases of vertex addition. In [16], Ma et al. researched the relation among the nullity, thedimension of cycle space and the number of pendant vertices of a graph.In recent years, the study of the rank and nullity of signed graphs received increased attention.The rank of signed planar graphs was investigated by Tian et al. [18]. Belardo et al. studiedthe Laplacian spectral of signed graphs in [1]. You et al. [14] characterized the nullity of signedgraphs. In [20], the relation between the rank of a signed graph and the rank of its underlyinggraph was researched by Wang. The nullity of unicyclic signed graphs and bicyclic signed graphswere studied by Fan et al. in [6] and [5], respectively. Wong et al. [21] characterized the positiveinertia index of the signed graphs. He et al. [9] studied the relation among the matching number,the cyclomatic number and the rank of the signed graphs. For other research of the rank of agraph one may be referred to those in [4, 12, 17, 11, 15].In this paper, the relation among the rank of a signed graph ( G, σ ) and the cyclomaticnumber and the independence number of its underlying graph is investigated. We prove that n − c ( G ) ≤ r ( G, σ ) + 2 α ( G ) ≤ n for any signed graph ( G, σ ) with order n . Moreover, theextremal graphs which attended the lower bound are characterized. Our main results are thefollowing Theorems 1.1 and 1.2. Theorem 1.1.
Let ( G, σ ) be a signed graph with order n . Then n − c ( G ) ≤ r ( G, σ ) + 2 α ( G ) ≤ n. Theorem 1.2.
Let ( G, σ ) be a signed graph with order n . Then r ( G, σ ) + 2 α ( G ) = 2 n − c ( G ) if and only if all the following conditions hold for ( G, σ ) : (i) the cycles (if any) of ( G, σ ) are pairwise vertex-disjoint; for each cycle (if any) C q of ( G, σ ) , either q ≡ and σ ( C q ) = + or q ≡ and σ ( C q ) = − ; (iii) α ( T G ) = α ([ T G ]) + c ( G ) . The rest of this paper is organized as follows. Prior to showing our main results, in Section2, some elementary notations and some useful lemmas are established. In Section 3, we give theproof of the main result of this paper. In Section 4, the extremal signed graphs which attainedthe lower bound of Theorem 1.1 are characterized.
In this section, some useful lemmas which will be used in the proofs of our main results arepresented.For X ⊆ V ( G ) , G − X is the induced subgraph obtained from G by deleting all vertices in X and all incident edges. In particular, G − { x } is usually written as G − x for simplicity. Foran induced subgraph H and a vertex u outside H , the induced subgraph of G with vertex set V ( H ) ∪ { u } is simply written as H + u . Lemma 2.1. [20]
Let ( G, σ ) be a signed graph. (i) If ( H, σ ) is an induced subgraph of ( G, σ ) , then r ( H, σ ) ≤ r ( G, σ ) . (ii) If ( G , σ ) , ( G , σ ) , · · · , ( G t , σ ) are all the connected components of ( G, σ ) , then r ( G, σ ) = P ti =1 r ( G i , σ ) . (iii) r ( G, σ ) ≥ with equality if and only if ( G, σ ) is an empty graph. Lemma 2.2. [6]
Let y be a pendant vertex of ( G, σ ) and x is the neighbour of y . Then r ( G, σ ) = r (( G, σ ) − { x, y } ) + 2 . Lemma 2.3. [2]
Let x be a vertex of ( G, σ ) . Then r ( G, σ ) − ≤ r (( G, σ ) − x ) ≤ r ( G, σ ) . Lemma 2.4. [4]
Let ( T, σ ) be a signed acyclic graph. Then r ( T, σ ) = r ( T ) = 2 m ( T ) . Lemma 2.5. [3]
Let T be a bipartite graph with order n . Then α ( T ) + m ( T ) = n . It is obviously that by Lemmas 2.4 and 2.5, we have
Lemma 2.6.
Let ( T, σ ) be a signed acyclic graph. Then r ( T, σ ) + 2 α ( T ) = 2 n . Lemma 2.7. [10]
Let y be a pendant vertex of a graph G and x is the neighbour of y . Then α ( G ) = α ( G − x ) = α ( G − { x, y } ) + 1 . Lemma 2.8. [14]
Let ( C n , σ ) be a signed cycle of order n . Then r ( C n , σ ) = n, if n is odd; n, if n ≡ and σ ( C n ) = − ; n, if n ≡ and σ ( C n ) = + ; n − , if n ≡ and σ ( C n ) = + ; n − , if n ≡ and σ ( C n ) = − . emma 2.9. [15] Let G be a graph with x ∈ V ( G ) . (i) c ( G ) = c ( G − x ) if x lies outside any cycle of G ; (ii) c ( G − x ) ≤ c ( G ) − if x lies on a cycle of G ; (iii) c ( G − x ) ≤ c ( G ) − if x is a common vertex of distinct cycles of G . Lemma 2.10. [10]
Let G be a graph. Then (i) α ( G ) − ≤ α ( G − x ) ≤ α ( G ) for any vertex x ∈ V ( G ) ; (ii) α ( G − e ) ≥ α ( G ) for any edge e ∈ E ( G ) . Lemma 2.11. [10]
Let T be a tree with at least one edge and T be the subtree obtained from T by deleting all pendant vertices of T . (i) α ( T ) ≤ α ( T ) + p ( T ) , where p ( T ) is the number of pendent vertices of T ; (ii) If α ( T ) = α ( T − D ) + | D | for a subset D of V ( T ) , then there is a pendant vertex x suchthat x / ∈ D . In this section, we study the relation among the rank and the independence number and thecyclomatic number of a signed graph, and give the proof for Theorem 1.1.
The proof of Theorem 1.1.
First, we prove the inequality on the left of Theorem 1.1. We argue by induction on c ( G ) toshow that n − c ( G ) ≤ r ( G, σ ) + 2 α ( G ) . If c ( G ) = 0 , then ( G, σ ) is a signed tree, and so resultfollows from Lemma 2.6. Hence we assume that c ( G ) ≥ . Let u be a vertex on some cycle of ( G, σ ) and ( G ′ , σ ) = ( G, σ ) − u . Let ( G , σ ) , ( G , σ ) , · · · , ( G l , σ ) be all connected components of ( G ′ , σ ) . By Lemma 2.9, we have l X i =1 c ( G i ) = c ( G ′ ) ≤ c ( G ) − . (1)By the induction hypothesis, one has n − − c ( G ′ ) ≤ r ( G ′ , σ ) + 2 α ( G ′ ) (2)By Lemmas 2.10 and 2.3, we have l X i =1 α ( G i ) = α ( G ′ ) ≤ α ( G ) , (3)and l X i =1 r ( G i , σ ) = r ( G ′ , σ ) ≤ r ( G, σ ) . (4)Thus the desired inequality now follows by combining (1), (2), (3) and (4), r ( G, σ ) + 2 α ( G ) ≥ r ( G ′ , σ ) + 2 α ( G ′ ) (5) ≥ n − − c ( G ′ ) ≥ n − − c ( G ) −
1) = 2 n − c ( G ) ,
4s desired.Next, we show that r ( G, σ ) + 2 α ( G ) ≤ n . Let I be a maximum independence set of G , i.e., | I | = α ( G ) . Then A ( G, σ ) = (cid:18) ⊤ A (cid:19) where B is a matrix of A ( G, σ ) with row indexed by I and column indexed by V ( G ) − I , B ⊤ refers to the transpose of B and A is the adjacency matrix of the induced subgraph G − I . Thenit can be checked that r ( G, σ ) ≤ r ( , B ) + r ( B ⊤ , A ) ≤ n − α ( G ) + n − α ( G ) = 2 n − α ( G ) . Thus, r ( G, σ ) + 2 α ( G ) ≤ n. This completes the proof of Theorem 1.1. (cid:3)
A signed graph ( G, σ ) with r ( G, σ ) + 2 α ( G ) = 2 n − c ( G ) is called a lower-optimal signedgraph. One can utilize the arguments above to make the following observations. Corollary 3.1.
Let u be a vertex of ( G, σ ) lying on a signed cycle. If r ( G, σ ) = 2 m ( G ) − c ( G ) ,then each of the following holds. (i) r ( G, σ ) = r (( G, σ ) − u ) ; (ii) ( G, σ ) − u is lower-optimal; (iii) c ( G ) = c ( G − u ) + 1 ; (iv) α ( G ) = α ( G − u ) ; (v) u lies on just one signed cycle of ( G, σ ) and u is not a quasi-pendant vertex of ( G, σ ) .Proof. In the proof arguments of Theorem 1.1 that justifies r ( G, σ ) + 2 α ( G ) ≥ n − c ( G ) . Ifboth ends of (5) are the same, then all inequalities in (5) must be equalities, and so Corollary3.1 (i)-(iv) are observed.To show (v). By Corollary 3.1 (iii) and Lemma 2.9, we conclude that u lies on just one signedcycle of ( G, σ ) . Suppose to the contrary that u is a quasi-pendant vertex which adjacent to apendant vertex v . Then by Lemma 2.2, we have r (( G, σ ) − u ) = r (( G, σ ) − { u, v } ) = r ( G, σ ) − , which is a contradiction to (i). This completes the proof of the corollary. Recall that a signed graph ( G, σ ) with order n is lower-optimal if r ( G, σ ) + 2 α ( G ) = 2 n − c ( G ) ,or equivalently, the signed graph which attain the lower bound in Theorem 1.1. In this section,we introduce some lemmas firstly, and then we give the proof for Theorem 1.2. By Lemma 2.8,the following Lemma 4.1 can be obtained directly. Lemma 4.1.
The signed cycle ( C q , σ ) is lower-optimal if and only if either q ≡ and σ ( C q ) = + or q ≡ and σ ( C q ) = − . emma 4.2. Let ( G, σ ) be a signed graph and ( G , σ ) , ( G , σ ) , · · · , ( G k , σ ) be all connectedcomponents of ( G, σ ) . Then ( G, σ ) is lower-optimal if and only if ( G j , σ ) is lower-optimal foreach j ∈ { , , · · · , k } .Proof. (Sufficiency.) For each i ∈ { , , · · · , k } , one has that r ( G i , σ ) + 2 α ( G i ) = 2 | V ( G i ) | − c ( G i ) . Then, one can get ( G, σ ) is lower-optimal immediately follows from the fact that r ( G, σ ) = k X j =1 r ( G j , σ ) ,α ( G ) = k X j =1 α ( G j ) , and c ( G ) = k X j =1 c ( G j ) . (Necessity.) Suppose to the contrary that there is a connected component of ( G, σ ) , say ( G , σ ) , which is not lower-optimal. Then r ( G , σ ) + 2 α ( G ) > | V ( G ) | − c ( G ) , and by Theorem 1.1, for each j ∈ { , , · · · , k } , we have r ( G j , σ ) + 2 α ( G j ) ≥ | V ( G j ) | − c ( G j ) . Thus, one has that r ( G, σ ) + 2 α ( G ) > | V ( G ) | − c ( G ) , a contradiction. Lemma 4.3.
Let u be a pendant vertex of a signed graph ( G, σ ) and v be the vertex whichadjacent to u . Let ( G , σ ) = ( G, σ ) − { u, v } . Then, ( G, σ ) is lower-optimal if and only if v isnot on any signed cycle of ( G, σ ) and ( G , σ ) is lower-optimal.Proof. (Sufficiency.) If v is not on any signed cycle, by Lemma 2.9, we have c ( G ) = c ( G ) . By Lemmas 2.2 and 2.7, one has that r ( G, σ ) = r ( G , σ ) + 2 , α ( G ) = α ( G ) + 1 . Thus, one can get ( G, σ ) is lower-optimal by the condition that r ( G , σ ) + 2 α ( G ) = 2 | V ( G ) | − c ( G ) . ( G, σ ) is lower-optimal,it can be checked that r ( G , σ ) + 2 α ( G ) = 2 | V ( G ) | − c ( G ) . It follows from Theorem 1.1 that one has r ( G , σ ) + 2 α ( G ) ≥ | V ( G ) | − c ( G ) . By the fact that c ( G ) ≤ c ( G ) , then we have c ( G ) = c ( G ) , r ( G , σ ) + 2 α ( G ) = 2 | V ( G ) | − c ( G ) . Thus ( G , σ ) is also lower-optimal and so the lemma is justified. Lemma 4.4.
Let ( G, σ ) be a signed graph obtained by joining a vertex x of a signed cycle C l bya signed edge to a vertex y of a signed connected graph ( K, σ ) . If ( G, σ ) is lower-optimal, thenthe following properties hold for ( G, σ ) . (i) Either l ≡ and σ ( C l ) = + or l ≡ and σ ( C l ) = − ; (ii) r ( G, σ ) = l − r ( K, σ ) , α ( G ) = l + α ( K ) ; (iii) ( K, σ ) is lower-optimal; (iv) Let ( G ′ , σ ) be the induced signed subgraph of ( G, σ ) with vertex set V ( K ) ∪ { x } . Then ( G ′ , σ ) is also lower-optimal; (v) α ( G ′ ) = α ( K ) + 1 and r ( G ′ , σ ) = r ( K, σ ) .Proof. (i): We show (i) by induction on the order n of ( G, σ ) . By Corollary 3.1, x can not be aquasi-pendant vertex of ( G, σ ) , then y is not an isolated vertex of ( G, σ ) . Then, ( K, σ ) containsat least two vertices, i.e., n ≥ l + 2 . If n = l + 2 , then ( K, σ ) contains exactly two vertices,without loss of generality, assume them be y and z . Thus, one has that ( C l , σ ) = ( G, σ ) − { y, z } .By Lemma 4.3, we have ( C l , σ ) is lower-optimal. Then (i) follows from Lemma 4.1 directly.Next, we consider the case of n ≥ l + 3 . Suppose that (i) holds for every lower-optimalsigned graph with order smaller than n . If ( K, σ ) is a forest. Then ( G, σ ) contains at least oneisolated vertex. Let u be a pendant vertex of ( G, σ ) and v be the vertex which adjacent to u . ByCorollary 3.1, v is not on ( C l , σ ) . By Lemma 4.3, one has that ( G, σ ) − { u, v } is lower-optimal.By induction hypothesis to ( G, σ ) − { u, v } , we have either l ≡ and σ ( C l ) = + or l ≡ and σ ( C l ) = − . If ( K, σ ) contains cycles. Let g be a vertex lying on a cycleof ( K, σ ) . By Corollary 3.1, ( G, σ ) − g is lower-optimal. Then, the induction hypothesis to ( G, σ ) − g implies that either l ≡ and σ ( C l ) = + or l ≡ and σ ( C l ) = − .This completes the proof of (i). (ii): Since x lies on a cycle of ( G, σ ) , by Corollary 3.1 and Lemmas 2.2 and 2.7, one has that r ( G, σ ) = r (( G, σ ) − x ) = r ( P l − , σ ) + r ( K, σ ) = l − r ( K, σ ) , (6)and α ( G ) = α ( G − x ) = α ( P l − ) + α ( K ) = l α ( K ) . (7) (iii): As C l is a pendant cycle of ( G, σ ) , one has that c ( K ) = c ( G ) − . (8)7y (6)-(8), we have r ( K, σ ) + 2 α ( K ) = 2( n − l ) − c ( K ) . (9) (iv): Let s be a vertex of ( C l , σ ) which adjacent to x . Then, by Corollary 3.1 and Lemmas2.2 and 2.7, we have r ( G, σ ) = r (( G, σ ) − s ) = l − r ( G ′ , σ ) , (10)and α ( G ) = α ( G − s ) = l −
22 + α ( G ′ ) . (11)Moreover,it is obviously that c ( G ) = c ( G ′ ) + 1 . (12)From (10)-(12), we have r ( G ′ , σ ) + 2 α ( G ′ ) = r ( G, σ ) + 2 α ( G ) − l − n − c ( G ) − l − n − l + 1) − c ( G ′ ) . (v): Combining (6) and (10), one has that r ( K, σ ) = r ( G ′ , σ ) . From (7) and (11), we have α ( K ) + 1 = α ( G ′ ) . This completes the proof.
Lemma 4.5.
Let ( G, σ ) be a lower-optimal signed graph. Then the following properties hold for ( G, σ ) . (i) The cycles (if any) of ( G, σ ) are pairwise vertex-disjoint; (ii) For each cycle C l of ( G, σ ) , either l ≡ and σ ( C l ) = + or l ≡ and σ ( C l ) = − ; (iii) α ( G ) = α ( T G ) + P C ∈ C G | V ( C ) | − c ( G ) .Proof. (i): By Corollary 3.1, (i) follows directly. (ii)-(iii):
We argue by induction on the order n of G to show (ii)-(iii). If n = 1 , then (ii)-(iii) hold trivially. Next, we consider the case of n ≥ . Suppose that (ii)-(iii) holds for everylower-optimal signed graph with order smaller than n .If E ( T G ) = 0 , i.e., T G is an empty graph, then each component of ( G, σ ) is a cycle or anisolated vertex. By Lemmas 4.1 and 4.2, we have either l ≡ and σ ( C l ) = + or l ≡ and σ ( C l ) = − . For each cycle C l , it is routine to check that α ( C l ) = ⌊ l ⌋ . Then(ii) and (iii) follows.If E ( T G ) ≥ . Then T G contains at least one pendant vertex, say x . If x is also a pendantvertex in ( G, σ ) , then ( G, σ ) contains a pendant vertex. If x is a vertex obtained by contractinga cycle of ( G, σ ) , then ( G, σ ) contains a pendant cycle. Then we will deal with the following twocases. 8 ase 1. If x is also a pendant vertex in ( G, σ ) . Let y be the unique neighbour of x and ( G , σ ) = ( G, σ ) − { x, y } . By Lemma 4.3, one has that y is not on any cycle of ( G, σ ) and ( G , σ ) is lower-optimal. By induction hypothesis, we have(a) For each cycle C l of ( G , σ ) , either l ≡ and σ ( C l ) = + or l ≡ and σ ( C l ) = − ;(b) α ( G ) = α ( T G ) + P C ∈ C G | V ( C ) | − c ( G ) .It is routine to check that all cycles of ( G, σ ) are also in ( G , σ ) . Then for each cycle C l of ( G, σ ) , either l ≡ and σ ( C l ) = + or l ≡ and σ ( C l ) = − . Thus (ii) follows.Furthermore, one has that c ( G ) = c ( G ) . Sine x is a pendant vertex of ( G, σ ) and y is a quasi-pendant vertex which is not in anycycle of ( G, σ ) , x is a pendant vertex of T G and y is a quasi-pendant vertex of T G . Moreover, T G = T G − { x, y } . Thus, by Lemma 2.7 and assertion (b), we have α ( G ) = α ( G ) + 1= α ( T G ) + X C ∈ C G | V ( C ) | − c ( G ) + 1= α ( T G ) − X C ∈ C G | V ( C ) | − c ( G ) + 1= α ( T G ) + X C ∈ C G | V ( C ) | − c ( G ) . Thus, (iii) holds in this case.
Case 2. ( G, σ ) contains a pendant cycle, say C q .In this case, let x be the unique vertex of C q of degree 3. Let K = G − C q and ( G , σ ) bethe induced signed subgraph of ( G, σ ) with vertex set V ( K ) ∪ { x } . By Lemma 4.4 (iv), one hasthat ( G , σ ) is lower-optimal. By induction hypothesis, we have(c) For each cycle C l of ( G , σ ) , either l ≡ and σ ( C l ) = + or l ≡ and σ ( C l ) = − ;(d) α ( G ) = α ( T G ) + P C ∈ C G | V ( C ) | − c ( G ) .By Lemma 4.4 (i), we have either q ≡ and σ ( C q ) = + or q ≡ and σ ( C q ) = − . Thus, (ii) follows from C G = C G ∪ C q = C K ∪ C q . Moreover, one has that X C ∈ C G | V ( C ) | X C ∈ C G | V ( C ) | q X C ∈ C K | V ( C ) | q . (13)Since C q is a pendant cycle of ( G, σ ) , it is obviously that c ( G ) = c ( K ) = c ( G ) − . (14)9y Lemma 4.4 (v), one has that α ( G ) = α ( K ) + 1 . (15)Note that T G = T G . (16)By Lemma 4.4 (ii) and (13)-(16), one has that α ( G ) = α ( K ) + p α ( G ) + p − α ( T G ) + X C ∈ C G | V ( C ) | − c ( G ) + p − α ( T G ) + X C ∈ C G | V ( C ) | − c ( G ) − α ( T G ) + X C ∈ C G | V ( C ) | − c ( G ) . This completes the proof.
The proof of Theorem 1.2. (Sufficiency.) We proceed by induction on the order n of ( G, σ ) .If n = 1 , then the result holds trivially. Therefore we assume that ( G, σ ) is a signed graph withorder n ≥ and satisfies (i)-(iii). Suppose that any signed graph of order smaller than n whichsatisfes (i)-(iii) is lower-optimal. Since the cycles (if any) of ( G, σ ) are pairwise vertex-disjoint, ( G, σ ) has exactly c ( G ) cycles, i.e., | O G | = c ( G ) .If E ( T G ) = 0 , i.e., T G is an empty graph, then each component of ( G, σ ) is a cycle or anisolated vertex. By (ii) and Lemma 4.1, we have ( G, σ ) is lower-optimal.If E ( T G ) ≥ . Then T G contains at least one pendant vertex. By (iii), one has that α ( T G ) = α ([ T G ]) + c ( G ) = α ( T G − O G ) + c ( G ) . Thus, by Lemma 2.11, there exists a pendent vertex of T G not in O G . Then, ( G, σ ) contains atleast one pendant vertex, say u . Let v be the unique neighbour of u and let ( G , σ ) = ( G, σ ) −{ u, v } . It is obviously that u is a pendant vertex of T G adjacent to v and T G = T G − { u, v } . ByLemma 2.7, one has that α ( T G ) = α ( T G − v ) = α ( T G − { u, v } ) + 1 . Claim. v does not lie on any cycle of ( G, σ ) .By contradiction, assume that v lies on a cycle of ( G, σ ) . Then v is in O G . Note that thesize of O G is c ( G ) . Then, H := ( T G − v ) ∪ K is a spanning subgraph of T G . Delete all theedges e in H such that e contains at least one end-vertex in O G \{ v } . Thus, the resulting graphis [ T G ] ∪ c ( G ) K . By Lemma 2.10, one has that α ([ T G ] ∪ c ( G ) K ) ≥ α (( T G − v ) ∪ K ) , α ([ T G ]) + c ( G ) ≥ α ( T G − v ) + 1 . Then, we have α ([ T G ]) ≥ α ( T G − v ) + 1 − c ( G ) = α ( T G ) + 1 − c ( G ) , a contradiction to (iii). This completes the proof of the claim.Thus, v does not lie on any cycle of ( G, σ ) . Moreover, u is also a pendant vertex of [ T G ] which adjacent to v and [ T G ] = T G − { u, v } . By Lemma 2.7, one has that α ([ T G ]) = α ([ T G ]) + 1 . It is routine to checked that c ( G ) = c ( G ) . Thus, α ( T G ) = α ( T G ) − α ([ T G ]) + c ( G ) − α ([ T G ]) + 1 + c ( G ) − α ([ T G ]) + c ( G ) . Combining the fact that all cycles of ( G, σ ) belong to ( G , σ ) , one has that ( G , σ ) satisfiesall the conditions (i)-(iii). By induction hypothesis, we have ( G , σ ) is lower-optimal. By Lemma4.3, we have ( G, σ ) is lower-optimal.(Necessity.) Let ( G, σ ) be a lower-optimal signed graph. If ( G, σ ) is a signed acyclic graph,then (i)-(iii) holds directly. So one can suppose that ( G, σ ) contains cycles. By Lemma 4.5, onehas that the cycles (if any) of ( G, σ ) are pairwise vertex-disjoint and for each cycle C l of ( G, σ ) ,either l ≡ and σ ( C l ) = + or l ≡ and σ ( C l ) = − . This completes the proofof (i) and (ii).Next, we argue by induction on the order n of ( G, σ ) to show (iii). Since ( G, σ ) containscycles, n ≥ . If n = 3 , then ( G, σ ) is a 3-cycle and (iii) holds trivially. Therefore we assumethat ( G, σ ) is a lower-optimal signed graph with order n ≥ . Suppose that (iii) holds for alllower-optimal signed graphs of order smaller than n .If E ( T G ) = 0 , i.e., T G is an empty graph, then each component of ( G, σ ) is a cycle or anisolated vertex. Then, (iii) follows immediately by Lemma 4.1.If E ( T G ) ≥ . Then T G contains at least one pendant vertex, say x . If x is also a pendantvertex in ( G, σ ) , then ( G, σ ) contains a pendant vertex. If x is a vertex obtained by contractinga cycle of ( G, σ ) , then ( G, σ ) contains a pendant cycle. Then we will deal with (iii) with thefollowing two cases. Case 1. x is a pendant vertex of ( G, σ ) .Let y be the unique neighbour of x and ( G , σ ) = ( G, σ ) − { x, y } . By Lemma 4.3, one hasthat y is not on any cycle of ( G, σ ) and ( G , σ ) is lower-optimal. By induction hypothesis, wehave α ( T G ) = α ([ T G ]) + c ( G ) .
11y the fact that y is not on any cycle of ( G, σ ) , then c ( G ) = c ( G ) . Note that x is also a pendant vertex of T G which adjacent to y , then T G = T G − { x, y } and [ T G ] = [ T G ] − { x, y } . By Lemma 2.7, one has that α ( T G ) = α ( T G ) + 1 , α ([ T G ]) = α ([ T G ]) + 1 . Thus, we have α ( T G ) = α ( T G ) + 1= α ([ T G ]) + c ( G ) + 1= α ([ T G ]) − c ( G ) + 1= α ([ T G ]) + c ( G ) . The result follows.
Case 2. ( G, σ ) contains a pendant cycle, say C q .Let u be the unique vertex of C q of degree 3 and K = G − C q . By Lemma 4.4, one has that ( K, σ ) is lower-optimal. By induction hypothesis, we have α ( T K ) = α ([ T K ]) + c ( K ) . (17)In view of Lemma 4.4, one has α ( G ) = α ( K ) + q . (18)It is obviously that C G = C K ∪ C q . Then, we have X C ∈ C G | V ( C ) | X C ∈ C K | V ( C ) | q . (19)Since ( G, σ ) and ( K, σ ) are lower-optimal, by Lemma 4.5, we have α ( T G ) = α ( G ) − X C ∈ C G | V ( C ) | c ( G ) , (20)and α ( T K ) = α ( K ) − X C ∈ C K | V ( C ) | c ( K ) . (21)Moreover, it is routine to check that c ( G ) = c ( K ) + 1 . (22)12ombining (17)-(22), we have α ( T G ) = α ( G ) − X C ∈ C G | V ( C ) | c ( G )= α ( K ) + q − X C ∈ C G | V ( C ) | c ( G )= α ( K ) − X C ∈ C K | V ( C ) | c ( G )= α ( K ) − X C ∈ C K | V ( C ) | c ( K ) + 1= α ( T K ) + 1 . Note that [ T G ] ∼ = [ T K ] . (23)Then, in view of (17) and (22)-(23), one has that α ( T G ) = α ( T K ) + 1= α ([ T K ]) + c ( K ) + 1= α ([ T G ]) + c ( G ) . This completes the proof. (cid:3)
Acknowledgments
This work was supported by the National Natural Science Foundation of China (No. 11731002,11771039 and 11771443), the Fundamental Research Funds for the Central Universities (No.2016JBZ012) and the 111 Project of China (B16002).