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Let G be a simple graph, and let G σ be an oriented graph of G with skew adjacency matrix S ( G σ ) . The skew spectral radius ρ s ( G σ ) of G σ is defined as the spectral radius of S ( G σ ) . When G is an odd-cycle graph (no even cycle), Cavers et al. (2012) [4] showed that the skew spectral radius of G σ is the same for every orientation σ of G . They proposed a problem: If G is a connected graph and ρ s ( G σ ) is the same for all orientations σ of G , must G be an odd-cycle graph? In this paper, we solve this problem and give a positive answer. [ABSTRACT FROM AUTHOR]

By a unified approach, we present sufficient conditions based on spectral radius for a graph to be k -connected, k -edge-connected, k -Hamiltonian, k -edge-Hamiltonian, β -deficient and k -path-coverable. [ABSTRACT FROM AUTHOR]

For a connected graph G , the distance signless Laplacian spectral radius of G is the spectral radius of its distance signless Laplacian matrix Q ( G ) defined as Q ( G ) = T r ( G ) + D ( G ) , where T r ( G ) is the diagonal matrix of vertex transmissions of G and D ( G ) is the distance matrix of G . In this paper, we study the effect of three types of graft transformations to decrease and/or increase the distance signless Laplacian spectral radius. As applications, on one hand, we determine the unique graphs with minimum distance signless Laplacian spectral radius among non-starlike trees, among non-caterpillar trees, and among non-starlike non-caterpillar trees, respectively, and determine the unique trees with third and forth minimum distance signless Laplacian spectral radius, respectively, and on the other hand, we determine the unique graphs with maximum distance signless Laplacian spectral radius among trees with given maximum degree, among trees of a perfect matching with maximum degree, among non-starlike trees, among non-caterpillar trees, among non-starlike non-caterpillar trees, and among unicyclic odd-cycle graphs, respectively, and determine the unique trees with second and third maximum distance signless Laplacian spectral radius, respectively, and the unique connected graph with second maximum distance signless Laplacian spectral radius. [ABSTRACT FROM AUTHOR]

In this paper, we give a sufficient condition on the spectral radius for a graph to be Hamiltonian. [ABSTRACT FROM AUTHOR]

The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. P. Hansen and D. Stevanović (2008) [9] determined the graphs with maximum spectral radius among all connected graphs of order n with diameter D . In this paper, we generalize this result to k -connected graphs of order n with diameter D . [ABSTRACT FROM AUTHOR]

In the paper, we will determine the graphs with maximal spectral radius among all the trees with n vertices and independence number α for ⌈ n 2 ⌉ ≤ α ≤ n − 1 . [ABSTRACT FROM AUTHOR]