A sharp upper bound on the spectral radius of θ(1,3,3)-free graphs with given size

2024/11/08 by Liu, Yuxiang, Wang, Ligong
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2411.05304

Abstract

A graph G is F-free if G does not contain F as a subgraph. Let ρ(G) be the spectral radius of a graph G. Let θ(1,p,q) denote the theta graph, which is obtained by connecting two distinct vertices with three internally disjoint paths with lengths 1, p, q, where p≤ q. Let S_n,k denote the graph obtained by joining every vertex of K_k to n-k isolated vertices and S_n,k^- denote the graph obtained from S_n,k by deleting an edge incident to a vertex of degree k, respectively. In this paper, we show that if ρ(G)≥ρ(S_(m+4)/(2),2^-) for a graph G with even size m≥ 92, then G contains a θ(1,3,3) unless G≅ S_(m+4)/(2),2^-.

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