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

The permanent of an n x n matrix M = (m(ij)) is defined as per(M) = P Qn s i=1 mis(i), where the sum is taken over all permutations s of {1,2,..., n}. The permanental polynomial of M, denoted by ik(M;x), is per(xIn M), 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 ik(L(G);x) and ik(Q(G);x) 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 ik(L(G);x), and (ii) the same signless Laplacian permanental polynomial as ik(Q(G);x), 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 S(r) n (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 S(r) n (for r = 2,3,4, 5), 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.