On the spectral Turán problem of theta graphs

Abstract

In 2010, Nikiforov conjectured that for $\ell \ge 2$ and n sufficiently large, $S_{n,\ell -1}^1$ is the unique graph with the maximum spectral radius over all n-vertex $C_{2\ell }$-free graphs. In 2022, Cioabǎ, Desai and Tait solved this conjecture. The theta graph ${\Theta }_{t,\ell }$ consists of two vertices joined by t vertex-disjoint paths, each of length ℓ. Particularly, ${\Theta }_{2,\ell }\cong C_{2\ell }$. In this paper, we characterize the unique extremal graph which attains the maximum spectral radius among all ${\Theta }_{t,\ell }$-free graphs of order n, where $t,\ell \ge 3$ and n is sufficiently large.