Extremal polygonal cacti for bond incident degree indices

For a connected graph G, the general sum-connectivity index χ _α (G), general Platt index Pl _α (G) and second Zagreb index M ₂ (G) are defined as χ _α (G)=∑ _uv∈E(G) (d(u)+d(v)) ^α , Pl _α (G)=∑ _uv∈E(G) (d(u)+d(v)−2) ^α and M ₂ (G)=∑ _uv∈E(G) d(u)d(v), where d(u) is the degree of vertex u. A k-polygonal cactus is a connected graph in which every edge lies on exactly one cycle of length k. In this paper, we shall determine the minimum value and maximum value for χ _α (G), Pl _α (G) and M ₂ (G), respectively, among the class of k-polygonal cacti with n cycles for k≥3, n≥3 and α>1. Furthermore, when α>1, we determine the unique extremal maximum k-polygonal cactus with n cycles for k≥3 and n≥3, and we also identify the unique extremal minimum k-polygonal cactus with n cycles for 3≤k≤5 and n≥3.