On signless Laplacian coefficients of bicyclic graphs

TL;DR

This paper investigates how certain transformations affect the signless Laplacian coefficients of bicyclic graphs, identifying specific graphs that minimize these coefficients within particular subclasses.

Contribution

It introduces transformations that reduce signless Laplacian coefficients and identifies extremal graphs with minimal coefficients in different bicyclic graph classes.

Findings

$B_n^1$ minimizes coefficients in $cal^1(n)$

$B_n^2$ minimizes coefficients in $cal^2(n)$

Transformations effectively decrease all signless Laplacian coefficients

Abstract

Let $G$ be a graph of order $n$ and $Q_G(x)= det(xI-Q(G))= \sum_{i=1}^n (-1)^i \varphi_i x^{n-i}$ 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 $\mathcal{B}(n)$ of all $n$-vertex bicyclic graphs. $\mathcal{B}^1(n)$ denotes all n-vertex bicyclic graphs with at least one odd cycle. We show that $B_n^1$ (obtained from $C_4$ 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 $\mathcal{B}^1(n)$. Moreover, we prove that $B_n^2$ (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 $\mathcal{B}^2(n)$ of all $n$-vertex bicyclic graphs with two even cycles.