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The  permanent  of an  n  ×  n  matrixis defined aswhere the sum is taken over all permutations  σ  ofThe  permanental polynomial  of  M , denoted byiswhere  I nis 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 polynomialsandare 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 asand ( ii ) the same signless Laplacian permanental polynomial asare 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.

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