On the signless Laplacian spectral radius of C_{4}-free k-cyclic graphs
aa r X i v : . [ m a t h . C O ] D ec On the signless Laplacian spectral radius of C -free k -cyclicgraphs ∗ Qi Kong , Ligong Wang Department of Applied Mathematics, School of Science, Northwestern Polytechnical University,Xi’an, Shaanxi 710072, People’s Republic of China. E-mail: [email protected] E-mail: [email protected]
Abstract A k -cyclic graph is a connected graph of order n and size n + k −
1. In thispaper, we determine the maximal signless Laplacian spectral radius and thecorresponding extremal graph among all C -free k -cyclic graphs of order n .Furthermore, we determine the first three unicyclic, and bicyclic, C -free graphswhose spectral radius of the signless Laplacian is maximal. Similar results areobtained for the (combinatorial) Laplacian. Key Words : k -cyclic graph; C -free; signless Laplacian spectral radius;Laplacian spectral radius. AMS Subject Classification (1991) : 05C50, 15A18.
Let G = ( V ( G ) , E ( G )) be a simple graph with vertex set V ( G ) and edge set E ( G ). Denote by v ( G ) the order of G and e ( G ) the size of G , that is to say, v ( G ) = | V ( G ) | , and e ( G ) = | E ( G ) | .Γ G ( u ) = { v | uv ∈ E ( G ) } and d G ( u ) = | Γ G ( u ) | , or simply Γ( u ) and d ( u ), respectively. Let δ = δ ( G ) and ∆ = ∆( G ) denote the minimum degree and maximum degree of the graph G . Let X and Y be disjoint subsets of V ( G ). e ( X, Y ) is the number of edges (in G ) joiningvertices in X to vertices in Y for G . Let P n , S n , C n and K n be the path, star, cycle andcomplete graph of order n , respectively.The union of G and G is the graph G ∪ G , whose vertex set is V ∪ V and whose edgeset is E ∪ E . kG denotes the union of k copies of G . The join of graphs G and G is thegraph G ∨ G obtained from G ∪ G by joining each vertex of G with every vertex of G .Let G kn = K ∨ ( kK ∪ ( n − k − K ) (see Fig. 1). We use U n to denote the family of allunicyclic graphs of order n , and B n to denote the family of all bicyclic graphs of order n .The matrix Q ( G ) = D ( G ) + A ( G ) is called the signless Laplacian matrix of G , where A ( G ) is the adjacency matrix of G and D ( G ) is the diagonal matrix of vertex degrees of G . ∗ Supported by the National Natural Science Foundation of China (No. 11171273) and sponsored by theSeed Foundation of Innovation and Creation for Graduate Students in Northwestern Polytechnical University(No. Z2016170) L ( G ) = D ( G ) − A ( G ) is called the Laplacian matrix of G . The largest eigenvalueof A ( G ), L ( G ) and Q ( G ) are called the spectral radius, Laplacian spectral radius and signlessLaplacian spectral radius (or Q -index) of G , respectively, and denoted by ρ ( G ), µ ( G ) and q ( G ), respectively.The central problem of the classical extremal graph theory is the Tur´an’s Problem:Problem A. Given a graph F , what is the maximum number of edges of a graph of order n , with no subgraph isomorphic to F ? Such problems are well understood nowadays, for example, see the book [2] and thesurvey [11]. Recently, Nikiforov, et al., investigated spectral Tur´an’ s Problem, namely forthe spectral radius ρ ( G ) of a graph G . In this new class of problems the central question isthe following one.Problem B. Given a graph F , what is the maximum spectral radius ρ ( G ) of a graph G oforder n , with no subgraph isomorphic to F ? In [11], when the graph F is the complete graph K r , the path or the cycle etc., Nikiforovdetermines the largest spectral radius of the graph G and their corresponding extremal graphs.The present paper contributes to an even newer trend in extremal graph theory, namely tothe study of variations of Promble A for the signless Laplacian spectral radius of graphs,where the central question is the following one.Problem C. Given a graph F , what is the maximum signless Laplacian spectral radius q ( G ) of a graph G of order n , with no subgraph isomorphic to F ? In [5], Nikiforov et al. determine the maximum signless Laplacian spectral radius of graphswith no 4-cycle and 5-cycle. And when the graph F is the complete graph K r , the path orthe cycle etc., the authors in [1, 12, 13, 14] determine the largest signless Laplacian spectralradius of the graph G and their corresponding extremal graphs, respectively. He and Guoin [8] determine the extremal graph of the signless Laplacian and Laplacian spectral radiusamong C -free k -cyclic graphs of order n .In this paper, we determine the signless Laplacian spectral radius of C -free k -cyclic graphsof order n and characterize its extremal graph. Furthermore, we determine the first threesignless Laplacian spectral radius of C -free unicyclic graphs of order n , and C -free bicyclicgraphs of order n with. In this section, we state some well-known results which will be used in this paper.
Lemma 2.1. ([9]) Let G be a connected graph of order n and q ( G ) its signless Laplacianspectral radius of G . Let u , v be two vertices of G and d ( v ) be the degree of vertex v . Assume v , v , . . . , v s (1 ≤ s ≤ d ( v )) are some vertices of Γ( v ) \ Γ( u ) and X = ( x , x , . . . , x n ) T is thePerron vector of Q ( G ) , where x i corresponds to the vertex v i (1 ≤ i ≤ n ) . Let G ∗ be the graphobtained from G by deleting the edges vv i and adding the edges uv i (1 ≤ i ≤ s ) , If x u ≥ x v ,then q ( G ) < q ( G ∗ ) . Lemma 2.2. ([4, 10]) For every graph G , we have q ( G ) ≤ max u ∈ V ( G ) { d ( u ) + 1 d ( u ) X v ∈ Γ( u ) d ( v ) } . If G is connected, equality holds if and only if G is regular or semiregular bipartite. emma 2.3. ([7, 3]) Let G be a simple connected graph on n vertices with maximum degree ∆ and at least one edge. Then µ ( G ) ≥ ∆( G ) + 1 , q ( G ) ≥ ∆( G ) + 1 ,where the former equality holds if and only if ∆( G ) = n − , and the latter one holds if andonly if G is the star S n . n v k v k v k v v v v v v Figure 1: The k -cyclic graph G kn v v v v n v v v v v v n v v v v v v n v v G G G v v Figure 2: The first three unicyclic graphs n v v v v v v v v G G G v v v n v v v v v v v n v v v v v v v v Figure 3: The first three bicyclic graphs
Lemma 2.4.
Let G kn , G and G be the graphs shown in Figs. 1 and 2. Then q ( G kn ) , q ( G ) and q ( G ) are the largest roots of the following equations, respectively, f k ( x ) , x − ( n + 3) x + 3 nx − k = 0 ,f ( x ) , x − ( n + 5) x + (6 n + 3) x − (9 n − x + (3 n + 8) x − ,f ( x ) , x − ( n + 5) x + (6 n + 4) x − (10 n − x + (3 n + 12) x − . Proof.
Let V ( G kn ) = { v , v , . . . , v n } and X = ( x , x , . . . , x n ) T be a Perron vectorcorrosponding to q ( G kn ). By the symmetry of G kn , we have3 = x = · · · = x k = x k +1 , x k +2 = · · · = x n .Let q = q ( G kn ), from the eigenequations for Q ( G kn ) we see that ( q − n + 1) x = 2 kx + ( n − k − x n , ( q − x = x , ( q − x n = x . Since X = ( x , x , . . . , x n ) T is an eigenvector corresponding to q ( G kn ), so X = 0. Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q − n + 1 − k − n + 2 k + 1 − q − − q − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . So q is the largest root of the following equation (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x − n + 1 − k − n + 2 k + 1 − x − − x − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . Consequently, q is the largest root of the following equation f k ( x ) , x − ( n + 3) x + 3 nx − k = 0.Using the same method, we obtain q ( G ) and q ( G ) are the largest roots of the followingequations, respectively, f ( x ) , x − ( n + 5) x + (6 n + 3) x − (9 n − x + (3 n + 8) x − .f ( x ) , x − ( n + 5) x + (6 n + 4) x − (10 n − x + (3 n + 12) x − . This completes the proof. (cid:3)
Let G be a C -free k -cyclic graph of order n . If k = 0 then G is a tree. In [6, 15], the authorsdetermined the first eight Laplacian spectral radius of trees of order n . For a bipartite graph G , by [3], we know that L ( G ) and Q ( G ) have the same eigenvalues. A tree is a bipartitegraph, so the results that are obtained by [6, 15] hold also for the signless Laplacian spectralradius of trees of order n . Therefore, in what follows, assume that k ≥ Theorem 3.1.
Let k ≥ , n ≥ k + 2 and let G be a C -free k -cyclic graph of order n . Then q ( G ) ≤ q ( G kn ) , with equality if and only if G = G kn , where q ( G kn ) is the largest root of the equation x − ( n + 3) x + 3 nx − k = 0 . Proof.
By Lemma 2.3, we have q ( G kn ) > ∆( G kn ) + 1 = n − n .4ince n ≥ k + 2 and G is a C -free k -cyclic graph of order n , we have ∆( G ) ≤ n −
1. If∆( G ) = n −
1, it is easy to see that G must be G kn . If G = G kn , then ∆( G ) ≤ n −
2. ByLemma 2.2, let w be a vertex of G such that d ( w ) + 1 d ( w ) X i ∈ Γ( w ) d ( i ) = max u ∈ V ( G ) { d ( u ) + 1 d ( u ) X v ∈ Γ( u ) d ( v ) } . Then q ( G ) ≤ d ( w ) + 1 d ( w ) X i ∈ Γ( w ) d ( i ) , and 1 ≤ d ( w ) ≤ ∆( G ) ≤ n − d ( w ) = 1 we obtain q ( G ) ≤ d ( w ) + 1 d ( w ) X i ∈ Γ( w ) d ( i ) ≤ ≤ n − < q ( G kn ) . If 2 ≤ d ( w ) ≤ n −
2, since G is C -free, every vertex v ∈ V ( G ) \ Γ( w ) has at most one neighborin Γ( w ). Then we have e (Γ( w ) , V ( G ) \ Γ( w )) ≤ d ( w ) + | V ( G ) \ (Γ( w ) ∪ { w } ) | = d ( w ) + n − d ( w ) − n − . Thus X i ∈ Γ( w ) d ( i ) ≤ d ( w ) + e (Γ( w ) , V ( G ) \ Γ( w )) ≤ d ( w ) + n − . Moreover, q ( G ) ≤ d ( w ) + 1 d ( w ) X i ∈ Γ( w ) d ( i ) ≤ d ( w ) + n − d ( w ) . Since the function f ( x ) = x + n − x is convex for x >
0, its maximum in any closed interval is attained at one of the ends of thisinterval. If 2 ≤ d ( w ) ≤ n −
2, then q ( G ) ≤ d ( w ) + 1 d ( w ) X i ∈ Γ( w ) d ( i ) ≤ { n − , n − n − n − } ≤ n ≤ q ( G kn ).From the above all, we obtain q ( G ) ≤ q ( G kn ), with equality if and only if G = G kn . Then fromLemma 2.4, we know that q ( G kn ) is the largest root of the polynomial x − ( n + 3) x + 3 nx − k = 0 . This completes the proof. (cid:3)
Theorem 3.2.
Let U n be the set unicyclic graphs of order n , and n ≥ , G i ∈ U n for i = 1 , , . Then for any G ∈ U n and G = G i ( i = 1 , , , we have q ( G ) < q ( G ) < q ( G ) < q ( G ) , where G , G and G are the graphs shown in Fig. 2. roof. For any G ∈ U n , from Theorem 3.1, if G = G , then q ( G ) < q ( G ). Especially, q ( G j ) < q ( G ) for j = 2 ,
3. Obviously, G and G are all the unicyclic graphs with ∆ = n − U n . Then for any G ∈ U n , if G = G i ( i = 1 , , G ) ≤ n −
3. By Lemma 2.3, q ( G j ) > ∆( G j ) + 1 = n − n − j = 2 ,
3. From Lemma 2.2, q ( G ) ≤ max u ∈ V ( G ) { d ( u ) + 1 d ( u ) X v ∈ Γ( u ) d ( v ) } . Similar to the proof of Theorem 3.1, we can get q ( G ) ≤ n − < q ( G i ).From Lemma 2.4, when x ≥ q ( G ) > n − n ≥
6, we have F ( x ) = f ( x ) − f ( x ) = x − ( n − x + 4 x = ( x − n + 1) + 2( n − x − n + 1) + ( n − n + 5)( x − n + 1) + 4( n − > f ( q ( G )) <
0, then q ( G ) > q ( G ).From the above all, for any G ∈ U n and G = G i , ( i = 1 , , q ( G ) < q ( G ) < q ( G ) < q ( G ) . This completes the proof. (cid:3)
Theorem 3.3.
Let B n be the set bicyclic graphs of order n , and n ≥ , G i ∈ B n for i = 4 , , .Then for any G ∈ B n and G = G i ( i = 4 , , , we have q ( G ) < q ( G ) < q ( G ) < q ( G ) , where G , G and G are shown in Fig. 3. Proof.
For any G ∈ B n , from Theorem 3.1, if G = G , then q ( G ) < q ( G ). Especially, q ( G j ) < q ( G )for j = 5 ,
6. Obviously, G and G are all the bicyclic graphs with ∆ = n − B n , then for any G ∈ B n , if G = G i ( i = 4 , , G ) ≤ n −
3. By Lemma 2.3, q ( G j ) > ∆( G j ) + 1 = n − n − j = 5 , q ( G ) ≤ max u ∈ V ( G ) { d ( u ) + 1 d ( u ) X v ∈ Γ( u ) d ( v ) } . Similar to the proof of Theorem 3.1, we can get q ( G ) ≤ n − < q ( G i ).Note that G = G − v v + v v = G − v v + v v . Applying Lemma 2.1 for the vertices v and v of the graph G , we have q ( G ) < q ( G ) . From the above all, for any G ∈ B n and G = G i , ( i = 4 , , q ( G ) < q ( G ) < q ( G ) < q ( G ) . This completes the proof. (cid:3)
Remark 3.4.
Lemma 2.3 is also true for the Laplacian special radius of graphs. From theproof of Theorem 3.1, we know that Theorem 3.1 also holds for Laplacian spectral radius ofgraphs. From [3], we know µ ( G ) ≤ q ( G ) , the equality holds if and only if G is a bipartitegraph. And from the proofs of Theorems 3.2 and 3.3, we know that Theorems 3.2 and 3.3also hold for Laplacian special radius of graphs. eferences [1] N.M.M. de Abreu and V. Nikiforov, Maxima of the Q-index: graphs with bounded cliquenumber, Electron. J. Linear Algebra , (2013) 121–130.[2] B. Bollob´as, Extremal Graph Theory, Academic Press, London
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