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GC
2008
Springer

Laplacian Spectrum of Weakly Quasi-threshold Graphs

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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...
R. B. Bapat, A. K. Lal, Sukanta Pati
Added 10 Dec 2010
Updated 10 Dec 2010
Type Journal
Year 2008
Where GC
Authors R. B. Bapat, A. K. Lal, Sukanta Pati
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