The spectral radius of graphs with no intersecting odd cycles

TL;DR

This paper characterizes the graphs with the largest spectral radius that do not contain a specific type of intersecting odd cycles, extending previous edge-count results to spectral properties.

Contribution

It determines the extremal graphs with maximum spectral radius avoiding certain intersecting odd cycles, advancing spectral extremal graph theory.

Findings

Identifies the graphs with maximum spectral radius avoiding $H_{s,t_1, dots,t_k}$.

Extends known extremal results from edge counts to spectral radius.

Provides structural characterization of extremal graphs for large n.

Abstract

Let $H_{s,t_1,\ldots ,t_k}$ be the graph with $s$ triangles and $k$ odd cycles of lengths $t_1,\ldots ,t_k\ge 5$ intersecting in exactly one common vertex. Recently, Hou, Qiu and Liu [Discrete Math. 341 (2018) 126--137], and Yuan [J. Graph Theory 89 (1) (2018) 26--39] determined independently the maximum number of edges in an $n$-vertex graph that does not contain $H_{s,t_1,\ldots ,t_k}$ as a subgraph. In this paper, we determine the graphs of order $n$ that attain the maximum spectral radius among all graphs containing no $H_{s,t_1,\ldots ,t_k}$ for $n$ large enough.