Spectral radius of graphs forbidden $C_7$ or $C_6^{\triangle}$

TL;DR

This paper determines the maximum spectral radius of graphs with a given size that do not contain a specific subgraph, and characterizes the structure of graphs with large spectral radius.

Contribution

It identifies the extremal graphs maximizing spectral radius under forbidden subgraph constraints and relates spectral radius bounds to cycle containment.

Findings

Identified the graph with maximum spectral radius avoiding $C_6^{\triangle}$.

Proved that high spectral radius implies the presence of all cycles $C_i$ for $3\le i\le 7$.

Characterized the structure of graphs with spectral radius at least $1+\sqrt{m-2}$.

Abstract

Let $C_k^{\triangle}$ be the graph obtained from a cycle $C_{k}$ by adding a new vertex connecting two adjacent vertices in $C_{k}$. In this note, we obtain the graph maximizing the spectral radius among all graphs with size $m$ and containing no subgraph isomorphic to $C_6^{\triangle}$. As a byproduct, we will show that if the spectral radius $\lambda(G)\ge1+\sqrt{m-2}$, then $G$ must contains all the cycles $C_i$ for $3\le i\le 7$ unless $G\cong K_3\nabla \left(\frac{m-3}{3}K_1\right)$.