Some bond incident degree indices of (molecular) graphs with fixed order and number of cut vertices
Summary: A bond incident degree (BID) index of a graph \(G\) is defined as \(\sum_{u v \in E (G)} f (d_G(u), d_G (v))\), 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. The choice \(f(d_G(u), d_G (v))= a^{d_G(u)}+ a^{d_G(v)}\) in the aforementioned formula gives the variable sum exdeg index \(\mathrm{SEI}_a\), where \(a\neq 1\) is any positive real number. A cut vertex of a graph \(G\) is a vertex whose removal results in a graph with more components than \(G\) has. A graph of maximum degree at most 4 is known as a molecular graph. Denote by \(\mathbb{V}_{n,k}\) the class of all \(n\)-vertex graphs with \(k\geq 1\) cut vertices and containing at least one cycle. Recently, \textit{J. Du} and \textit{X. Sun}, characterized the graphs having the maximum value of \(\mathrm{SEI}_a\) from the set \(\mathbb{V}_n^k\) for \(a>1\) [``On the graph connectivity and the variable sum exdeg index'', AIMS Math. 6, No. 1, 607--622 (2021; \url{doi:10.3934/math.2021037})]. In the present paper, we not only characterize the graphs with the minimum value of \(\mathrm{SEI}_a\) from the set \(\mathbb{V}_n^k\) for \(a>1\), but we also solve a more general problem concerning a special type of BID indices. As the obtained extremal graphs are molecular graphs, they remain extremal if one considers the class of all \(n\)-vertex molecular graphs with \(k\geq 1\) cut vertices and containing at least one cycle.
scientific article; zbMATH DE number 7443992
scientific article