Abstract
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 ∑ u v ∈ E G f d G u , d G v , where d G w denotes the degree of a vertex w of G , E G 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: I f i G = ∑ u v ∈ E G f i d G u / d G u + f i d G v / d G v , where i ∈ 1,2 , f 1 is a strictly convex function, and f 2 is a strictly concave function. More precisely, graphs attaining the minimum and maximum I f i 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.
Highlights
All the graphs considered in this study are connected. e notation and terminology that are used in this study but not defined here can be found in some standard graph-theoretical books [6, 7].Graph invariants of the following form are known as the bond incident degree (BID) indices [5]: BID(G) f dG(u), dG(v), (1) uv∈E(G)where dG(w) denotes the degree of a vertex w ∈ V(G) of the graph G, E(G) is the edge set of G, and f is a real-valued symmetric function
We are concerned with the following type [2] of the BID indices: Ifi(G)
We study the graphs attaining the minimum and maximum Ifi values from the class of all cactus graphs with a given number of vertices and cycles
Summary
Akbar Ali ,1 Akhlaq Ahmad Bhatti, Naveed Iqbal ,1 Tariq Alraqad ,1 Jaya Percival Mazorodze ,3 Hicham Saber, and Abdulaziz M. A connected graph in which no edge lies on more than one cycle is called a cactus graph ( known as Husimi tree). A bond incident degree (BID) index of a graph G is defined as uv∈E(G)f(dG(u), dG(v)), where dG(w) denotes the degree of a vertex w of G, E(G) is the edge set of G, and f is a real-valued symmetric function. Is study involves extremal results of cactus graphs concerning the following type of the BID indices: Ifi(G) uv∈E(G)[fi(dG(u))/dG(u) + fi(dG(v))/dG(v)], where i ∈ {1, 2}, f1 is a strictly convex function, and f2 is a strictly concave function. 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. 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. e obtained results cover several well-known indices including the general zeroth-order Randicindex, multiplicative first and second Zagreb indices, and variable sum exdeg index
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