Laplacian Spectrum of Weakly Quasi-threshold Graphs

In this paper we study the class of weakly quasi-threshold graphs that are obtained from a vertex by recursively applying the operations (i) adding a new isolated vertex, (ii) adding a new vertex and making it adjacent to all old vertices, (iii) disjoint union of two old graphs, and (iv) adding a new vertex and making it adjacent to all neighbours of an old vertex. This class contains the class of quasi-threshold graphs. We show that weakly quasi-threshold graphs are precisely the comparability graphs of a forest consisting of rooted trees with each vertex of a tree being replaced by an independent set. We also supply a quadratic time algorithm in the the size of the vertex set for recognizing such a graph. We completely determine the Laplacian spectrum of weakly quasi-threshold graphs. It turns out that weakly quasi-threshold graphs are Laplacian integral. As a corollary we obtain a closed formula for the number of spanning trees in such graphs. A conjecture of Grone and Merris assert...

• Type: Journal
• Year: 2008
• Where: GC
• Authors: R. B. Bapat, A. K. Lal, Sukanta Pati
• Keywords: Applied Computing, Comparability Graphs, GC 2008, Laplacian, Quasi-threshold Graphs