Spectral Turán problems for intersecting even cycles

Abstract

Let C2k1,2k2,…,2kt denote the graph obtained by intersecting t distinct even cycles C2k1,C2k2,…,C2kt at a unique vertex. In this paper, we determine the unique graph with maximum adjacency spectral radius among all graphs on n vertices that do not contain any C2k1,2k2,…,2kt as a subgraph, for n sufficiently large. When one of the constituent even cycles is a C4, our results improve upper bounds on the Turán numbers for intersecting even cycles that follow from more general results of Füredi [21] and Alon, Krivelevich and Sudakov [1]. Our results may be seen as extensions of previous results for spectral Turán problems on forbidden even cycles C2k,k≥2 (see [8,36,46,47]).