Spectral radius of graphs of given size with forbidden subgraphs

Abstract

Let ρ(G) be the spectral radius of a graph G with m edges. Let S k m−k+1 be the graph obtained from K 1,m−k by adding k disjoint edges within its independent set. Nosal's theorem states that if ρ(G) > √ m, then G contains a triangle. Zhai and Shu showed that any non-bipartite graph G with m ≥ 26 and ρDiscrete Math. 345 (2022) 112630]. Wang proved that if ρ(G) ≥ √ m − 1 for a graph G with size m ≥ 27, then G contains a quadrilateral unless G is one of four exceptional graphs [Z.W. Wang, Discrete Math. 345 (2022) 112973]. In this paper, we show that any non-bipartite graph G with size m ≥ 51 and ρ(G) ≥ ρ(S 2 m−1 ) > √ m − 2 contains a quadrilateral unless G is one of three exceptional graphs. Moreover, we show that if ρ(G) ≥ ρ(S − m+4 2 ,2 ) for a graph G with even size m ≥ 74, then G contains a C + 5 unless G ∼ = S − m+4 2 ,2 , where C + t denotes the graph obtained from C t and C 3 by identifying an edge, S n,k denotes the graph obtained by joining each vertex of K k to n − k isolated vertices and S − n,k denotes the graph obtained by deleting an edge incident to a vertex of degree two, respectively.

A Spectral Extremal Problem on Non-Bipartite Triangle-Free Graphs

“…Inequality (3) impulsed the great interests of studying the maximum spectral radius for F -free graphs with given number of edges, see [32,35] for K r+1 -free graphs, [34,50,46] for C 4 -free graphs, [49] for K 2,r+1 -free graphs, [49,31] for C 5 -free or C 6 -free graphs, [29] for C 7 -free graphs, [43,9,26] for C △ 4 -free or C △ 5 -free graphs, where C △ k is a graph on k + 1 vertices obtained from C k and C 3 by sharing a common edge; see [37] for B k -free graphs, where B k denotes the book graph consisting of k triangles sharing a common edge, [21] for F 2 -free graphs with given number of edges, where F 2 is the friendship graph consisting of two triangles intersecting in a common vertex, [38,39] for counting the number of C 3 and C 4 . We refer the readers to the surveys [36,18] and references therein.…”

“…For example, the C 5 -free or C 6 -free spectral extremal graphs with m edges are determined in [49] when m is odd, and later in [31] when m is even. Moreover, the C △ 4 -free or C △ 5 -free spectral extremal graphs are determined in [43] for odd m, and subsequently in [9,26] for even m. In addition, the results of Nikiforov [33], Zhai and Wang [48] showed that the C 4 -free spectral extremal graphs with given order n also rely on the parity of n. In a nutshell, for large size m, there is a common phenomenon that the extremal graphs in two cases are extremely similar, that is, the extremal graph in the even case is always constructed from that in the odd case by handing an edge to a vertex with maximum degree. Surprisingly, the extremal graphs in our conclusion break down this common phenomenon and show a new structure of the extremal graphs.…”