A spectral condition for odd cycles in non-bipartite graphs

Let A(G) be the adjacency matrix of a graph G and rho(G) be its spectral radius. Given a graph Hand a family Fof graphs, let ex(sp)(n, H; F) = max{rho(G)parallel to V(G)vertical bar = n, H subset of G, G does not contain any graph of F}. Let S2k-1(K-s,K-t) be the graph obtained by replacing an edge of K-s,K-t with a copy of P2k+1, where k >= 2. In this paper, we show that ex(sp)(n, C2k+3; {C-3, C-5,..., C2k+1}) =rho(S2k-1(Kinverted right perpendicularn-2k+1/2inverted left perpendicular, (left perpendicularn-2k+1/2inverted right perpendicular))) and the unique extremal graph is S2k-1((inverted right perpendicularn-2k+1/2inverted left perpendicular) (left perpendicularn-2k+1/2inverted right perpendicular,) ), which solves a question proposed in [Eigenvalues and triangles in graphs, Comb. Probab. Comput. 30 (2021) 258-270]. (c) 2021 Elsevier Inc. All rights reserved.

Automated summary

This paper demonstrates extremal cases under certain conditions and resolves a previously proposed question.