On signless Laplacian coefficients of bicyclic graphs

2012/12/20 by Zhang, Jie, Zhang, Xiao-Dong
#05C50 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1212.5261

Abstract

Let G be a graph of order n and Q_G(x)= det(xI-Q(G))= ∑_i=1ⁿ (-1)ⁱ φ_i x^n-i be the characteristic polynomial of the signless Laplacian matrix of a graph G. We give some transformations of G which decrease all signless Laplacian coefficients in the set B(n) of all n-vertex bicyclic graphs. B¹(n) denotes all n-vertex bicyclic graphs with at least one odd cycle. We show that B_n¹ (obtained from C₄ by adding one edge between two non-adjacent vertices and adding n-4 pendent vertices at the vertex of degree 3) minimizes all the signless Laplacian coefficients in the set B¹(n). Moreover, we prove that B_n² (obtained from K_2,3 by adding n-5 pendent vertices at one vertex of degree 3) has minimum signless Laplacian coefficients in the set B²(n) of all n-vertex bicyclic graphs with two even cycles.

Related

• On signless Laplacian coefficients of unicyclic graphs with given matching number
• The characterizing properties of (signless) Laplacian permanental polynomials of bicyclic graphs
• Signless laplacian characteristic polynomials of regular graph transformations
• Laplacian coefficients of unicyclic graphs with the number of leaves and girth
• Bicyclic graphs with exactly two main signless Laplacian eigenvalues