Signless Laplacian polynomial for splice and link of graphs

Abstract

The signless Laplacian matrix of a graph G is Q(G) = A(G) + D(G), where A(G) is the adjacency matrix and D(G) is the diagonal degree matrix of a graph G. The characteristic polynomial of the signless Laplacian matrix is called the signless Laplacian polynomial. The present work is all about the study of signless Laplacian polynomial for the splice of more than two graphs and the link of such graphs. It is noted that such a study is easier when we take into account of the vertex set partition being an equitable partition, because equitable partition of the vertex set reduces the computational steps and also the quotient matrix polynomial is a part of the polynomial of a graph. In this paper we consider the splice and links of complete graphs and of complete bipartite graphs and obtain the signless Laplacian polynomial of these using equitable partition of the vertex set.