%0 Generic %1 ai2020proximity %A Ai, Jiangdong %A Gerke, Stefanie %A Gutin, Gregory %A Mafunda, Sonwabile %D 2020 %K 4330 graph_seminar network_analysis proximity %T Proximity and Remoteness in Directed and Undirected Graphs %U http://arxiv.org/abs/2001.10253 %X Let $D$ be a strongly connected digraph. The average distance $\sigma(v)$ of a vertex $v$ of $D$ is the arithmetic mean of the distances from $v$ to all other vertices of $D$. The remoteness $\rho(D)$ and proximity $\pi(D)$ of $D$ are the maximum and the minimum of the average distances of the vertices of $D$, respectively. We obtain sharp upper and lower bounds on $\pi(D)$ and $\rho(D)$ as a function of the order $n$ of $D$ and describe the extreme digraphs for all the bounds. We also obtain such bounds for strong tournaments. We show that for a strong tournament $T$, we have $\pi(T)=\rho(T)$ if and only if $T$ is regular. Due to this result, one may conjecture that every strong digraph $D$ with $\pi(D)=\rho(D)$ is regular. We present an infinite family of non-regular strong digraphs $D$ such that $\pi(D)=\rho(D).$ We describe such a family for undirected graphs as well.

@misc{ai2020proximity, abstract = {Let $D$ be a strongly connected digraph. The average distance $\bar{\sigma}(v)$ of a vertex $v$ of $D$ is the arithmetic mean of the distances from $v$ to all other vertices of $D$. The remoteness $\rho(D)$ and proximity $\pi(D)$ of $D$ are the maximum and the minimum of the average distances of the vertices of $D$, respectively. We obtain sharp upper and lower bounds on $\pi(D)$ and $\rho(D)$ as a function of the order $n$ of $D$ and describe the extreme digraphs for all the bounds. We also obtain such bounds for strong tournaments. We show that for a strong tournament $T$, we have $\pi(T)=\rho(T)$ if and only if $T$ is regular. Due to this result, one may conjecture that every strong digraph $D$ with $\pi(D)=\rho(D)$ is regular. We present an infinite family of non-regular strong digraphs $D$ such that $\pi(D)=\rho(D).$ We describe such a family for undirected graphs as well.}, added-at = {2020-01-29T16:14:58.000+0100}, author = {Ai, Jiangdong and Gerke, Stefanie and Gutin, Gregory and Mafunda, Sonwabile}, biburl = {https://www.bibsonomy.org/bibtex/25e874fab2633881e810320b1db779b95/j.c.m.janssen}, description = {Proximity and Remoteness in Directed and Undirected Graphs}, interhash = {ca64c5c08b2d89f0f9e2657c31e9826f}, intrahash = {5e874fab2633881e810320b1db779b95}, keywords = {4330 graph_seminar network_analysis proximity}, note = {cite arxiv:2001.10253}, timestamp = {2020-01-29T16:14:58.000+0100}, title = {Proximity and Remoteness in Directed and Undirected Graphs}, url = {http://arxiv.org/abs/2001.10253}, year = 2020 }