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 $$\beta \le \frac{n}{4}$$ .