On the general degree-eccentricity index of a graph

Abstract

We define the general degree-eccentricity index of a connected graph G as $DEI_{a,b}(G) = \sum _{v \in V(G)} d_{G}^{a}(v) ecc_{G}^{b}(v)$ for $a,b \in {\mathbb {R}}$, where V(G) is the vertex set of G, $d_{G} (v)$ is the degree of a vertex v and $ecc_{G}(v)$ is the eccentricity of v in G. Let $I_1$ be any index which decreases with the addition of edges and let $I_2$ be any index which increases with the addition of edges. We obtain sharp lower bounds on the $I_1$ index and the $DEI_{a,b}$ index, where $a < 0$ and $b > 0$, and sharp upper bounds on the $I_2$ index and the $DEI_{a,b}$ index, where $a > 0$ and $b < 0$, for connected graphs of given order in combination with given independence number, vertex cover number or minimum degree. We also present sharp upper bounds on the $DEI_{a,b}$ index, where $a \ge 1$ and $b < 0$, for connected graphs of given order n in combination with given vertex connectivity, edge connectivity, number of pendant vertices, number of bridges or matching number $\beta \le \frac{n}{4}$.