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A 2-edge-connected graph G is minimally 2-edge-connected if deleting arbitrary edge of G always leaves a graph which is not 2-edge-connected. In this paper, we completely characterize the (minimally) 2-edge-connected graphs with given size having the maximal signless Laplacian (Laplacian) index. [ABSTRACT FROM AUTHOR]

The A α -matrix of a graph G was defined by Nikiforov in 2017 as A α (G) = α D (G) + (1 − α) A (G) , where α ∈ [ 0 , 1 ] , D (G) and A (G) are the diagonal matrix of degrees and the adjacency matrix respectively. The largest eigenvalue of A α (G) is called A α -index of G. In this paper, we completely determine the extremal graphs with maximal A α -index among all graphs with size m , domination number γ and no isolated vertices for α ∈ [ 1 2 , 1). [ABSTRACT FROM AUTHOR]

Nikiforov defined the A α -matrix of a graph G as A α (G) = α D (G) + (1 − α) A (G) , where α ∈ [ 0 , 1 ] , D (G) and A (G) are the diagonal matrix of degrees and the adjacency matrix respectively. The largest eigenvalue of A α (G) is called the A α -spectral radius of G , denoted by ρ α (G). In this paper, we first give an upper bound on ρ α (G) of a connected graph G with fixed size m ≥ 3 k and maximum degree Δ ≤ m − k , where k is a positive integer. For two connected graphs G 1 and G 2 with size m ≥ 4 , employing this upper bound, we prove that ρ α (G 1) > ρ α (G 2) if Δ (G 1) > Δ (G 2) and Δ (G 1) ≥ 2 m 3 + 1. As an application, we determine the graph with the maximal A α -spectral radius among all graphs with fixed size and girth. Our theorems generalize the recent results for the signless Laplacian spectral radius of a graph. [ABSTRACT FROM AUTHOR]

A 3-connected graph G is minimally 3-connected if it becomes not 3-connected after deleting any edge of G. In this paper, we give an upper bound on the signless Laplacian spectral radius of minimally 3-connected graphs with m edges, and completely characterize the corresponding extremal graph in the case when m is even. As a corollary, we determine the unique graph with the maximal signless Laplacian spectral radius among all Halin graphs with m edges. [ABSTRACT FROM AUTHOR]

In this paper, we study the signless Laplacian spectral Turán type problem in terms of the size, obtain sharp upper bounds on the signless spectral radius of C 3 ( C 4 )-free graphs with given size and minimum degree δ ≥ 2 , and characterize the corresponding extremal graphs completely. [ABSTRACT FROM AUTHOR]

Let G be a simple undirected graph. For real number α ∈ [0, 1], Nikiforov defined the Aα-matrix of G as Aα(G) = αD(G) + (1 − α)A(G), where A(G) and D(G) are the adjacency matrix and the degree diagonal matrix of G respectively. In this paper, we obtain a sharp upper bound on the largest eigenvalue ρα(G) of Aα(G) for α ∈ [1/2, 1). Employing this upper bound, we prove that 'For connected G1 and G2 with n vertices and m edges, if the maximum degree Δ(G1) ≥ 2α(1 − α)(2m − n + 1) + 2α and Δ(G1) > Δ(G2), then ρ α (G 1) > ρ α (G 2) '. Let λα(G) denote the least eigenvalue of Aα(G). For α ∈ (1/2, 1), we prove that 'For two connected G1 and G2, if the minimum degree δ (G 1) ≤ 1 1 − α − 2 and δ(G1) < δ(G2), then λα(G1) < λα(G2)'. [ABSTRACT FROM AUTHOR]

A 2-connected graph G is minimally 2-connected if deleting any arbitrary chosen edge of G always leaves a graph which is not 2-connected. In this paper, we give sharp upper bounds for the Laplacian spectral radius of (minimally) 2-connected graphs with given size, and determine the corresponding extremal graphs. [ABSTRACT FROM AUTHOR]

A 2-connected graph G is minimally 2-connected if deleting any arbitrary chosen edge of G always leaves a graph which is not 2-connected. In this paper, we give sharp upper bounds on the Q -index of (minimally) 2-connected graphs with given size, and characterize the corresponding extremal graphs completely. [ABSTRACT FROM AUTHOR]

Let q(G) denote the signless Laplacian spectral radius of a graph G. In this paper, we first give an upper bound on q(G) of a connected graph G with fixed size m ≥ 3 k (k ∈ Z +) and maximum degree Δ ≤ m - k . For two connected graphs G 1 and G 2 with size m ≥ 4 , employing this upper bound, we prove that q (G 1) > q (G 2) if Δ (G 1) > Δ (G 2) and Δ (G 1) ≥ 2 m 3 + 1 . As an application, we determine the first ⌊ d / 2 ⌋ graphs with the largest signless Laplacian spectral radius among all graphs with fixed size and diameter. [ABSTRACT FROM AUTHOR]

Let g (G) denote the girth of a graph G. In this paper, we determine the unique graph with the maximum least Q-eigenvalue among all unicyclic graphs of order n = 6k (k ≥ 8) with perfect matchings. For the cases when n = 6k + 2 and n = 6k + 4, we prove that g (G) = 3 if G is a graph with the maximum least Q-eigenvalue, and provide a conjecture and a problem on the sharp upper bound of the least Q-eigenvalue. [ABSTRACT FROM AUTHOR]

Let G be a simple undirected graph. For any real number α ∈ [ 0 , 1 ] , Nikiforov defined the A α -matrix of G as A α (G) = α D (G) + (1 - α) A (G) , where A(G) and D(G) are the adjacency matrix and the degree diagonal matrix of G respectively. The largest eigenvalue of A α (G) is called the A α -spectral radius of G. In this paper, we give sharp upper bounds on the A α -spectral radius of C 4 -free graphs and Halin graphs for α ∈ [ 1 / 2 , 1) respectively. [ABSTRACT FROM AUTHOR]

A single-cone tree (unicyclic graph) is the join of a complete graph K 1 and a tree (unicyclic graph). Suppose π = (d 1 , d 2 , ... , d n) and π ′ = (d 1 ′ , d 2 ′ , ... , d n ′) are two non-increasing degree sequences. We say π is majorizated by π ′ , denoted by π ⊲ π ′ , if and only if π ≠ π ′ , ∑ i = 1 n d i = ∑ i = 1 n d i ′ , and ∑ i = 1 j d i ≤ ∑ i = 1 j d i ′ for all j = 1 , 2 , ... , n - 1 . We use J π to denote the class of single-cone trees (unicyclic graphs) with degree sequence π . Suppose that π and π ′ are two different non-increasing degree sequences of single-cone trees (unicyclic graphs). Let ρ and ρ ′ be the largest spectral radius of the graphs in J π and J π ′ , respectively, μ and μ ′ be the largest signless Laplacian spectral radius of the graphs in J π and J π ′ , respectively. In this paper, we prove that if π ⊲ π ′ , then ρ < ρ ′ and μ < μ ′ . [ABSTRACT FROM AUTHOR]

The Laplacian spread of a graph G is defined to be the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of G. Let Q t (n , d) = { G | G - v 0 is a tree on n - 1 vertices and d G (v 0) = d }. Recently, Y. Xu and J. Meng characterized the unique graph with maximum Laplacian spread among all graphs in the set Q t (n , d) with 1 ≤ d ≤ (n - 4) / 2 . In this paper, we extend their result by determining the unique graph with maximum Laplacian spread among all graphs in the set Q t (n , d) with 1 ≤ d ≤ n - 5 . [ABSTRACT FROM AUTHOR]

In this paper, we study the algebraic connectivity of a Hamiltonian graph, and determine all Hamiltonian graphs whose algebraic connectivity attain the minimum among all Hamiltonian graphs onnvertices. [ABSTRACT FROM AUTHOR]

In this paper, we study the least Q-eigenvalue of a non-bipartite hamiltonian graph, and determine all non-bipartite hamiltonian graphs whose least Q-eigenvalue attains the minimum among all non-bipartite hamiltonian graphs on n vertices. [ABSTRACT FROM AUTHOR]

In this paper, we determine the unique graph whose least signless Laplacian eigenvalue attains the minimum among all non-bipartite unicyclic graphs of order n with maximum degree Δ and among all non-bipartite connected graphs of order n with maximum degree Δ, respectively. [ABSTRACT FROM AUTHOR]

A unicyclic graph is a connected graph whose number of edges is equal to the number of vertices. Fan et al. (Discrete Math. 313:903-909, ) and Liu et al. (Electron. J. Linear Algebra 26:333-344, ) determined, independently, the unique unicyclic graph whose least Q-eigenvalue attains the minimum among all non-bipartite unicyclic graphs of order n with k pendant vertices. In this paper, we extend their results and determine the first three non-bipartite unicyclic graphs of order n with k pendant vertices ordering by least Q-eigenvalue. [ABSTRACT FROM AUTHOR]

Let G be a simple and connected graph of order n (≥ 6). The eigenvalues of the signless Laplacian matrix Q (G) of G are called the Q -eigenvalues of G. Let q n (G) be the least Q -eigenvalue of G. In this paper, we investigate the multiplicity of q n (G) and characterize all n -vertex connected graphs with q n (G) of multiplicity n − 3. Moreover, it is shown that all these graphs are determined by their signless Laplacian spectra. [ABSTRACT FROM AUTHOR]

TheQ-indexof graphGis the largest eigenvalueq(G)of its signless Laplacian. In this paper, we prove that the graphhas the maximalQ-index among all outer-planar graphs of ordern. [ABSTRACT FROM PUBLISHER]

In this paper, for a nonbipartite graph with both order n and domination number γ , we show that it contains a nonbipartite unicyclic spanning subgraph with the same domination number γ . We also present some results about the changing of the domination number when the structure of a graph is changed. By investigating the relation between the domination number and the least Q -eigenvalue of a graph, we minimize the least Q -eigenvalue among all the nonbipartite graphs with both order n ⩾ 4 and given domination number γ ⩽ n + 1 3 . [ABSTRACT FROM AUTHOR]

Abstract: Let G be a simple connected graph on n vertices and m edges. Lima et al. (2011) in [2] posed the following conjecture on the least eigenvalue of the signless Laplacian of G: . In this paper we prove a stronger result: For any graph with n vertices and m edges, we have . [Copyright &y& Elsevier]

A c-cyclic graph is a connected graph with n vertices and n + c−1 edges. In this paper, we consider the problem that: in the class of graphs, each of which is the complement graph of a c-cyclic graph with n vertices, which graph has the largest A α -spectral radius. We shall show that, for 0 ≤ c ≤ n − 4 , the extremal graph must be a graph with an isolated vertex. This implies that the maximum degree of G is larger, then the A α -spectral radius of its complement graph is also larger. We also determine the structure of the extremal graph with largest A α -spectral radius among the class of graphs, each of which is the complement graph of a c-cyclic graph with n vertices and maximum degree n−1 for 0 ≤ c ≤ n − 3 and n−2 for 0 ≤ c ≤ n − 4 , respectively. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove that if and are subsets of a finite abelian group with , then , where and. In addition, we give a complete description of the subsets and of such that and . Our results generalize the corresponding theorems of Gallardo et al. in cyclic group [L. Gallardo, G. Grekos, L. Habsieger, et al., Restricted addition in and an application to the Erdös–Ginzburg–Ziv problem, J. London Math. Soc. 65 (2) (2002) 513–523]. [Copyright &y& Elsevier]

Abstract: Tricyclic graphs are connected graphs in which the number of edges equals the number of vertices plus two. In this paper, we determine the graph with the largest Laplacian spectral radius among all tricyclic graphs with vertices and pendant vertices. [Copyright &y& Elsevier]

Abstract: Denote by the set of trees with n vertices and matching number q. Guo [On the Laplacian spectral radius of a tree, Linear Algebra Appl. 368 (2003) 379–385] gave the tree in with the greatest value of the largest Laplacian eigenvalue. In this paper, we give another proof of this result. Using our method, we can go further beyond Guo by giving the tree in with the second largest value of the largest Laplacian eigenvalue. [Copyright &y& Elsevier]

Abstract: Bicyclic graphs are connected graphs in which the number of edges equals the number of vertices plus one. In this paper we determine the graph with the largest spectral radius among all bicyclic graphs with n vertices and diameter d. As an application, we give first three graphs among all bicyclic graphs on n vertices, ordered according to their spectral radii in decreasing order. [Copyright &y& Elsevier]

Abstract: In this paper, we determine graphs with the largest spectral radius among all the unicyclic and all the bicyclic graphs with n vertices and k pendant vertices, respectively. [Copyright &y& Elsevier]

Abstract: Connected graphs in which the number of edges equals the number of vertices plus one are called bicyclic graphs. In this paper, all bicyclic graphs whose second largest eigenvalue does not exceed 1 have been determined. [Copyright &y& Elsevier]

Abstract: A rational cuboid is a rectangular parallelepiped whose edges and face diagonals all have rational lengths. In this paper, we consider the problem: are there rational cuboids with a given face? In a sense, we reduce the problem to a finite calculation. [Copyright &y& Elsevier]

Let A (G) and D (G) be the adjacency matrix and the degree diagonal matrix of a graph G , respectively. For any real number α ∈ [ 0 , 1 ] , Nikiforov defined the A α -matrix of a graph G as A α (G) = α D (G) + (1 − α) A (G). The A α -spread of a graph is the difference between the largest eigenvalue and the smallest eigenvalue of the A α -matrix of the graph. In this paper, we obtain some new lower bounds on A α -spread and characterize the extremal graphs for certain cases, which extend the results of A -spread and Q -spread. Moreover, some graph operations on A α -spread are presented. [ABSTRACT FROM AUTHOR]

Abstract Denote by L g, l the lollipop graph obtained by attaching a pendant path P = v g v g + 1 ⋯ v g + l (l ≥ 1) to a cycle C = v 1 v 2 ⋯ v g v 1 (g ≥ 3). A F g , l − g r a p h of order n ≥ g + 1 is defined to be the graph obtained by attaching n − g − l pendent vertices to some of the nonpendant vertices of L g, l in which each vertex other than v g + l − 1 is attached at most one pendant vertex. A F g , l ∘ -graph is a F g , l − g r a p h in which v g is attached with pendant vertex. Denote by q min the least Q − e i g e n v a l u e of a graph. In this paper, we proceed on considering the domination number, the least Q -eigenvalue of a graph as well as their relation. Further results obtained are as follows: (i) some results about the changing of the domination number under the structural perturbation of a graph are represented; (ii) among all nonbipartite unicyclic graphs of order n , with both domination number γ and girth g (g ≤ n − 1), the minimum q min attains at a F g , l -graph for some l ; (iii) among the nonbipartite graphs of order n and with given domination number which contain a F g , l ∘ -graph as a subgraph, some lower bounds for q min are represented; (iv) among the nonbipartite graphs of order n and with given domination number n 2. [ABSTRACT FROM AUTHOR]