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A k-tree is a spanning tree in which every vertex has degree at most k. In this paper, we provide a sufficient condition for the existence of a k-tree in a connected graph with fixed order in terms of the adjacency spectral radius and the signless Laplacian spectral radius, respectively. Also, we give a similar condition for the existence of a perfect matching in a balanced bipartite graph with fixed order and minimum degree. [ABSTRACT FROM AUTHOR]

In this paper, all connected graphs with three distinct signless Laplacian eigenvalues of which the second one is at most five are determined. [ABSTRACT FROM AUTHOR]

Denote by (resp. ) the distance (resp. distance signless Laplacian) eigenvalues of a connected graph G on n vertices, and by (resp. ) the multiplicity of (resp. ). In this paper, we completely determine the graphs with and , respectively. [ABSTRACT FROM AUTHOR]

A graph is called Laplacian integral if all its Laplacian eigenvalues are integers. In this paper, we give an edge subdividing theorem for Laplacian eigenvalues of a graph (Theorem 2.1) and characterize a class ofk-cyclic graphs whose algebraic connectivity is less than one. Using these results, we determine all the Laplacian integral tricyclic graphs. Furthermore, we show that all the Laplacian integral tricyclic graphs are determined by their Laplacian spectra. [ABSTRACT FROM AUTHOR]