Spectral extremal problem on the square of ℓ-cycle

Abstract

Let Cℓ be the cycle of order ℓ. The square of Cℓ, denoted by Cℓ2, is obtained by joining all pairs of vertices with distance no more than two in Cℓ. Denote by ex(n,F) and spex(n,F) the maximum size and maximum spectral radius over all n-vertex F-free graphs, respectively. The well-known Turán problem asks for ex(n,F), and Nikiforov in 2010 proposed a spectral counterpart, known as the Brualdi-Solheid-Turán type problem, focusing on determining spex(n,F). In this paper, for any integer ℓ≥6 that is not divisible by 3, we characterize the unique extremal graph with respect to ex(n,Cℓ2) and spex(n,Cℓ2) for sufficiently large n, respectively.