A study on determination of some graphs by Laplacian and signless Laplacian permanental polynomials

Abstract

The permanent of an n × n matrix is defined as where the sum is taken over all permutations σ of The permanental polynomial of M, denoted by is where In is the identity matrix of order n. Let G be a simple undirected graph on n vertices and its Laplacian and signless Laplacian matrices be L(G) and Q(G) respectively. The permanental polynomials and are called the Laplacian permanental polynomial and signless Laplacian permanental polynomial of G respectively. A graph G is said to be determined by its (signless) Laplacian permanental polynomial if all the graphs having the same (signless) Laplacian permanental polynomial with G are isomorphic to G. A graph G is said to be combinedly determined by its Laplacian and signless Laplacian permanental polynomials if all the graphs having (i) the same Laplacian permanental polynomial as and (ii) the same signless Laplacian permanental polynomial as are isomorphic to G. In this article we investigate the determination of some graphs, namely, star, wheel, friendship graphs and a particular kind of caterpillar graph (whose all r non-pendant vertices have the same degree n) by their Laplacian and signless Laplacian permanental polynomials. We show that a kind of caterpillar graphs (for ), wheel graph (up to 7 vertices) and friendship graph (up to 7 vertices) are determined by their (signless) Laplacian permanental polynomials. Further we prove that all friendship graphs and wheel graphs are combinedly determined by their Laplacian and signless Laplacian permanental polynomials.