Groupe d’études et de recherche en analyse des décisions

# Maximum eccentric connectivity index for graphs with given diameter

## Pierre Hauweele, Alain Hertz, Hadrien Mélot, Bernard Ries et Gauvain Devillez

The eccentricity of a vertex $$v$$ in a graph $$G$$ is the maximum distance between $$v$$ and any other vertex of $$G$$. The diameter of a graph $$G$$ is the maximum eccentricity of a vertex in $$G$$. The eccentric connectivity index of a connected graph is the sum over all vertices of the product between eccentricity and degree. Given two integers $$n$$ and $$D$$ with $$D\leq n-1$$, we characterize those graphs which have the largest eccentric connectivity index among all connected graphs of order $$n$$ and diameter $$D$$. As a corollary, we also characterize those graphs which have the largest eccentric connectivity index among all connected graphs of a given order $$n$$.

, 14 pages