On unicyclic graphs with a given girth and their minimum symmetric division deg index

The degree of a vertex u in a graph G is denoted by dG(u).The symmetric division deg (SDD) index of G is denoted by SDD(G) and is defined as SDD(G) = xy∈E(G) dG(x)/dG(y) + dG(y)/dG(x), where E(G) is the set of all edges of G.A connected graph with the same number of vertices and edges is known as a unicyclic graph.The girth of a unicyclic graph is the number of edges of its unique cycle.This paper solves the problem of characterizing graphs attaining the first two minimum values of the SDD index over the class of all unicyclic graphs of fixed order and with a given girth.Applications of the obtained results yield the solution to the problem of determining graphs having the first three minimum values of the SDD index among all unicyclic graphs of a given order.