Spanning Trees of Bounded Degree, Connectivity, Toughness, and the Spectrum of a Graph

Recently, Cioabǎ and Gu obtained a relationship between the spectrum of a regular graph and the existence of spanning trees of bounded degree, generalized connectivity and toughness, respectively. In this paper, motivated by the idea of Cioabǎ and Gu, we determine a connection between the (signless Laplacian and Laplacian) eigenvalues of a graph and its structural properties involving the existence of spanning trees with bounded degrees and generalized connectivity, respectively. We also present a connection between the (signless Laplacian and Laplacian) eigenvalues and toughness of a bipartite graph, respectively. Finally, we obtain a lower bound of toughness in a graph in terms of edge connectivity κ^′ and maximum degree Δ.