The maximum spectral radius of non-bipartite graphs forbidding short odd cycles

Abstract

It is well-known that eigenvalues of graphs can be used to describe structural properties and parameters of graphs. A theorem of Nosal states that if G is a triangle-free graph with m edges, then λ(G) ≤ √ m, equality holds if and only if G is a complete bipartite graph. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)] proved a generalization for non-bipartite trianglefree graphs. Moreover, Zhai and Shu [Discrete Math. 345 (2022)] presented a further improvement. In this paper, we present an alternative method for proving the improvement by Zhai and Shu. Furthermore, the method can allow us to give a refinement on the result of Zhai and Shu for non-bipartite graphs without short odd cycles.

Refinement on spectral Turán's theorem

“…Our result is not only a refinement on the spectral Turán theorem, but it is also a spectral version of Brouwer's theorem. In a forthcoming paper [31], we shall present some extensions and generalizations on Nosal's theorem for graphs with given number of edges.…”

“…Over the past few years, various extensions and generalizations on Nosal's theorem have been obtained in the literature; see, e.g., [36,37,38,51] for extensions of K r+1free graphs, [33,54,55,31] for extensions of graphs with given size. In addition, many spectral extremal problems are also obtained recently; see [11,12] for the friendship graph and the odd wheel, [29,13] for intersecting odd cycles and cliques, [50] for a recent conjecture.…”

Spectral radius of graphs of given size with forbidden subgraphs

“…Combining with(19), we haveρ * x w = v∈N (w) x v ≤ v∈N 0 (u * ) x v < 1 2 x u * , it follows that x w < x u * 2ρ * < x u * 18. By(12), we havee(W ) < e(N (u * )) − |N + (u * )contradiction. Thus W = ∅.…”

“…Wang [21] showed that if ρ(G) ≥ √ m − 2 for a non-bipartite graph G of size m ≥ 26, then G contains a triangle unless G is one of some exceptional graphs. For more details, one may refer to [9,12,13] and references therein.…”