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: \[ RDD_*=um_{ubstack{\{u,v\}ubseteq V(G) ueq v}}\frac{ẹg(u)\cdotẹg(v)}{peratorname{dist}(u,v)} \] where $\deg(v)$ is the degree of the vertex $v$, and $\operatorname{dist}(u,v)$ 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.