New bounds on proximity and remoteness in graphs
The average distance of a vertex $v$ of a connected graph $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The proximity $\pi(G)$ and the remoteness $\rho(G)$ of $G$ are defined as the minimum and maximum, respectively, average distance of the vert...
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| Format: | Article |
| Language: | English |
| Published: |
Azarbaijan Shahide Madani University
2016-01-01
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| Series: | Communications in Combinatorics and Optimization |
| Subjects: | |
| Online Access: | http://comb-opt.azaruniv.ac.ir/article_13543.html |
| Summary: | The average distance of a vertex $v$ of a connected graph $G$
is the arithmetic mean of the distances from $v$ to all
other vertices of $G$. The proximity $\pi(G)$ and the remoteness $\rho(G)$
of $G$ are defined as the minimum and maximum, respectively, average
distance of the vertices of $G$.
In this paper we investigate the difference between proximity or remoteness and the classical distance
parameters diameter and radius.
Among other results we show that in a graph of order
$n$ and minimum degree $\delta$ the difference between
diameter and proximity and the difference between
radius and proximity cannot exceed
$\frac{9n}{4(\delta+1)}+c_1$ and
$\frac{3n}{4(\delta+1)}+c_2$, respectively, for
constants $c_1$ and $c_2$ which depend on $\delta$
but not on $n$. These bounds improve bounds by
Aouchiche and Hansen \cite{AouHan2011} in terms of
order alone by about a factor of $\frac{3}{\delta+1}$.
We further give lower bounds on the remoteness in
terms of diameter or radius. Finally we show that
the average distance of a graph, i.e., the average of
the distances between all pairs of vertices, cannot
exceed twice the proximity. |
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| ISSN: | 2538-2128 2538-2136 |