General Degree-Eccentricity Index of Trees

Abstract

For a connected graph G and $a,b \in \mathbb {R}$, the general degree-eccentricity index is defined as $\mathrm{DEI}_{a,b}(G) = \sum _{v \in V(G)} d_{G}^{a}(v) \mathrm{ecc}_{G}^{b}(v)$, where V(G) is the vertex set of G, $d_{G} (v)$ is the degree of a vertex v and $\mathrm{ecc}_{G}(v)$ is the eccentricity of v in G. We obtain sharp upper and lower bounds on the general degree-eccentricity index for trees of given order in combination with given matching number, independence number, domination number or bipartition. The bounds hold for $0< a < 1$ and $b > 0$, or for $a > 1$ and $b < 0$. Many bounds hold also for $a = 1$. All the extremal graphs are presented.