On The Spectral Radius And A Sharp Lower Bound Of Randi(?) Index Of Cacti With Perfect Matchings

Let G = (V, E) be a simple and connected graph with the vertex set V(G) and the edge set E(G). Letλ₁;λ₂;…;λ_n are all n eigenvalues of the characteristic polynomial det(λ_I- A(G)) of G, where the largest eigenvalue is called the spectral radius of G. The Randic index of the graph is one of the most important topological indices in chemical graph theory. It has a lot of applications in chemistry and been widely investigated as well. And the Randic index of the graph G is defined as R(G) = sum from uv∈E(G) (d(u)d(v))^-1/2, where d(u) and d(v) are the degrees of vertices u andv in G, respectively.We say that the graph G is a cactus if any two of its cycles have at most one common vertex. The set of cacti with k cycles and perfect matchings on 2n vertices are denoted by (?)(n, k) and the Randic index of a graph G is denoted by R(G). In this paper, the graph with largest spectral radius is uniquely characterized and a sharp lower bound of index of Randic is also given among G∈(?)(n,k): If G∈(?) (n, k)\ {H₆, H₈}, n≥2, then R(G)≥n+k-1/(?)+1/(?)+n-1/(?)+(?)k/2, where H₆, H₈ are displayed in Figure 3-1.