Spectral extremal problem for intersecting one even and one odd cycle

Abstract

Let G be a graph of size m and $\rho \left(G\right)$ be the spectral radius of its adjacency matrix. Let $H_k$ be the graph obtained from k triangles and a quadrilateral sharing a common vertex. The graph $H_1$ is also called the fish graph. Let $S_{n,2}$ be the graph obtained by joining each vertex of $K_2$ to $n-2$ isolated vertices. Li and Peng (2022) solved the spectral Turán problem for intersecting odd cycles and Desai (2024) determined the unique graph with maximum spectral radius among all graphs that do not contain intersecting even cycles, for n sufficiently large. Further, Li, Lu and Peng (2023) determined the unique m-edge graph with maximum spectral radius forbidding two intersecting triangles at a common vertex. Now, a natural question that arises is determining the maximum spectral radius for graphs that avoid intersecting cycles, with at least one of odd length and at least one of even length. In this paper, we address this question by investigating the spectral extremal problem for the graph formed by the intersection of an even cycle and an odd cycle at a common vertex. We show that if G is an $H_1$-free graph of size $m\ge 44$, then $\rho \left(G\right)\le \frac{1+\sqrt{4m-3}}{2}$, with equality if and only if $G\cong S_{\frac{m+3}{2},2}$.