Permanental Bounds for the Signless Laplacian Matrix of a Unicyclic Graph with Diameter d

Abstract

Let U n be the set of n-vertex unicyclic graphs, U d n be the set of n-vertex unicyclic graphs of diameter d. In this paper we determine the second-minimum value of signless Laplacian permanent of graphs among U n ; as well we obtain the lower bound for the signless Laplacian permanent of graphs in U d n . The corresponding extremal graphs are characterized.

The characterizing properties of (signless) Laplacian permanental polynomials of bicyclic graphs

“…on the signless Laplacian permanental polynomials. Li and Zhang [11,12] gave the bounds of constant terms of signless Laplacian permanental polynomials of some graphs. Liu [17] showed that complete graphs and complete bipartite graphs are determined by their signless Laplacian permanental polynomials.…”

The Characterizing Properties of (Signless) Laplacian Permanental Polynomials of Almost Complete Graphs

“…We call π(L(G), x) (resp, π(Q(G), x)) the Laplacian (resp, signless Laplacian) permanental polynomial of G. e Laplacian permanental polynomial of a graph was first considered by Merris et al [2], and the signless Laplacian permanental polynomial was first studied by Faria [3]. For more studies on (signless) Laplacian permanental polynomials, see [4][5][6][7][8][9][10][11][12][13][14], among others.…”

Edge-grafting theorems on permanents of Laplacian matrices of graphs and their applications

“…For more recent results on Laplacian (resp. signless Laplacian) permanent one is referred to [5,11,12].…”