### Reciprocal Product-Degree Distance of Graphs

#### Abstract

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.

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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