A new upper bound for the spectral radius of graphs with girth at least 5

Let G be a simple connected graph with V(G) = {v1, v2, … , vn} and girth at least 5. Let Δ be the maximum degree of G. In this paper, we present a new sharp upper bound for the spectral radius of G as follows:ρ(G)⩽-1+4n+4Δ-32.Moreover, equality holds if and only if G ≅ C5.