Spectral conditions for the existence of specified paths and cycles in graphs

TL;DR

This paper establishes sharp spectral bounds on the least eigenvalue of graphs to determine the existence of specific paths and cycles, providing new spectral conditions for graph structure analysis.

Contribution

It introduces new sharp bounds on the least eigenvalue for graphs lacking certain paths or cycles and identifies extremal graphs, advancing spectral graph theory.

Findings

Sharp bounds on the least eigenvalue for graphs without specific paths or cycles

Characterization of extremal graphs achieving these bounds

Spectral conditions for the existence of particular paths and cycles

Abstract

Let $G$ be a graph with $n$ vertices and $\lambda_n(G)$ be the least eigenvalue of its adjacency matrix of $G$. In this paper, we give sharp bounds on the least eigenvalue of graphs without given pathes or cycles and determine the extremal graphs. This result gives spectral conditions for the existence of specified paths and cycles in graphs.