Spectral Turan problem of non-bipartite graphs: Forbidden books

A book graph Br+1 is a set of r+1 triangles with a common edge, where r >= 0 is an integer. Zhai and Lin (2023) proved that for n >= 13/2 r, if G is a Br+1-free graph of order n, then rho(G) <= rho(T-n,T-2), with equality if and only if G congruent to T-n,T-2. Note that the extremal graph T-n,T-2 is bipartite. Motivated by the above elegant result, we investigate the spectral Turan problem of non-bipartite Br+1-free graphs of order n. For r = 0, Lin et al. (2021) provided a complete solution and proved a nice result: If G is a non-bipartite trianglefree graph of order n, then rho(G) <= rho(SKleft perpndicular (n-1/2) (right perpndicular), (inverted) (left perpndicular) n-1/2 (inverted) (right perpndicular) ), with equality if and only if G congruent to SKleft perpndicular (n-1/2) (right perpndicular), (inverted) (left perpndicular) n-1/2 (inverted) (right perpndicular,) where SK (left perpndicular) (n-1/2) (right perpndicular), (inverted) (left perpndicular) n-1/2 (inverted) (right perpndicular) is the graph obtained from K-left perpndicular (n-1/2) (right perpndicular), (inverted) (left perpndicular) n-1/2 (inverted) (right perpndicular) by subdividing an edge. For general r >= 1, let K-left perpndicular (n-1/2) (right perpndicular), (inverted) (left perpndicular) n-1/2 (inverted) (r,r)(right perpndicular) be the graph obtained from K-left perpndicular (n-1/2) (right perpndicular), (inverted) (left perpndicular) n-1/2 (inverted) (right perpndicular) by adding a new vertex v(0) such that v(0) has exactly r neighbors in each part of K-left perpndicular (n-1/2) (right perpndicular), (inverted) (left perpndicular) n-1/2 (inverted) (right perpndicular). By adopting adifferent technique named the residual index, Chvatal-Hanson theorem and typical spectral extremal methods, we in this paper prove that: If G is a non-bipartite Br+1-free graph of order n, then rho(G) <= rho (K-left perpndicular (n-1/2) (right perpndicular), (inverted) (left perpndicular) n-1/2 (inverted) (r,r)(right perpndicular)), with equality if and only if G congruent to K-left perpndicular (n-1/2) (right perpndicular), (inverted) (left perpndicular) n-1/2 (inverted) (r,r)(right perpndicular) extremal graphs are completely different for r = 0 and general r >= 1. (c) 2025 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.