Maximum Eccentric Connectivity Index for Graphs with Given Diameter
Pierre Hauweele, Alain Hertz, Hadrien Mélot, Bernard Ries, Gauvain Devillez
TL;DR
The paper determines the exact maximal eccentric connectivity index $\xi^c(G)$ for connected graphs with given order $n$ and (i) fixed diameter $D$ and (ii) no fixed diameter. It introduces the extremal graph families ${\sf M}_n$ (near-complete graphs) and ${\sf E}_{n,D,k}$ ( Path-plus-clique constructions) and derives explicit formulas for $\xi^c$ on these graphs. For $D\ge 3$, the maximum equals $f(n,D)$ and is achieved by a specific subfamily $\mathcal{C}_n^D$ (with two small exceptions), while for $D=2$ the maximum is achieved by ${\sf M}_n$ with a precise parity-based bound. When diameter is not fixed, the extremal graphs are ${\sf M}_n$ for small $n$, but for large $n$ the graphs ${\sf E}_{n,D^*,n-D^*-1}$ with $D^*={\lceil(n+1)/3\rceil}+1$ maximize $\xi^c$, with a detailed dependence on $n\bmod 6$. The work culminates in a table of optimal graphs and raises a conjecture for the case of fixed order and size, conjecturing that a specific ${\sf E}_{n,D,k}$ attains the maximum in general.
Abstract
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 $n$.