On the Minimum General Sum-Connectivity of Trees of Fixed Order and Pendent Vertices

Abstract

For a graph $G$ , its general sum-connectivity is usually denoted by ${\chi }_{\alpha }\left(G\right)$ and is defined as the sum of the numbers ${\left[d_G\left(u\right)+d_G\left(v\right)\right]}^{\alpha }$ over all edges $uv$ of $G$ , where $d_G\left(u\right),d_G\left(v\right)$ represent degrees of the vertices $u,v$ , respectively, and $\alpha$ is a real number. This paper addresses the problem of finding graphs possessing the minimum ${\chi }_{\alpha }$ value over the class of all trees with a fixed order $n$ and fixed number of pendent vertices $n_1$ for $\alpha >1$ . This problem is solved here for the case when $4\le n_1\le \left(n+5\right)/3$ and $\alpha >1$ , by deriving a lower bound on ${\chi }_{\alpha }$ for trees in terms of their orders and number of pendent vertices.