Spectral radius of graphs forbidden C7 or [formula omitted

Abstract

Let Ck△ be the graph obtained from a cycle Ck by adding a new vertex connecting two adjacent vertices in Ck. In this note, we obtain the graph maximizing the spectral radius among all graphs with size m and containing no subgraph isomorphic to C6△. As a byproduct, we will show that if the spectral radius λ(G)≥1+m−2, then G must contain all the cycles Ci for 3≤i≤7 unless G≅K3∇(m−33K1).