Reciprocal Product-Degree Distance of Graphs

Guifu Su, Liming Xiong, Ivan Gutman, Lan Xu


We investigate a new graph invariant named reciprocal
product--degree distance, defined as:
RDD_* =\sum_{\stackrel{\{u,v\}\subseteq V(G)}{u \neq v}}
\frac{\deg(u)\cdot \deg(v)}{\mathrm{dist}(u,v)}
where $\deg(v)$ is the degree of the vertex $v$, and
$\mathrm{dist}(u,v)$ is the distance between the vertices $u$ and
$v$ in the underlying graph. $RDD_{*}$ is a product--degree
modification of the Harary index. We determine the connected
graph of given order with maximum $RDD_{*}$-value, and establish
lower and upper bounds for $RDD_{*}$. Also a Nordhaus--Gaddum--type
relation for $RDD_{*}$ is obtained.

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