Some Bond Incident Degree Indices of Cactus Graphs - Akbar Ali; Akhlaq Ahmad Bhatti; Naveed Iqbal; Tariq Alraqad; Jaya Percival Mazorodze; Hicham Saber; Abdulaziz M. Alanazi (Journal of Mathematics)
A connected graph in which no edge lies on more than one cycle is called a cactus graph (also known as Husimi tree). A bond incident degree (BID) index of a graph G is defined as ∑uv∈EGfdGu,dGv, where dGw denotes the degree of a vertex w of G, EG is the edge set of G, and f is a real-valued symmetric function. This study involves extremal results of cactus graphs concerning the following type of the BID indices: IfiG=∑uv∈EGfidGu/dGu+fidGv/dGv, where i∈1,2, f1 is a strictly convex function, and f2 is a strictly concave function. More precisely, graphs attaining the minimum and maximum Ifi values are studied in the class of all cactus graphs with a given number of vertices and cycles. The obtained results cover several well-known indices including the general zeroth-order Randić index, multiplicative first and second Zagreb indices, and variable sum exdeg index.