On $\ell$-distance balanced product graphs

TL;DR

This paper characterizes and analyzes $ ext{ell}$-distance-balanced properties in various graph products, providing new conditions, correcting previous assertions, and extending known results for specific graph classes.

Contribution

It offers a complete characterization of $ ext{ell}$-distance-balanced corona and lexicographic products, and extends known results for $1$-distance-balanced graphs to general $ ext{ell}$-distance-balanced graphs.

Findings

Characterization of $ ext{ell}$-distance-balanced corona products

Characterization of lexicographic products for $ ext{ell} \,\geq 3$

Conditions under which $K_n \,\square\, H$ is $ ext{ell}$-distance-balanced

Abstract

A graph $G$ is $\ell$-distance-balanced if for each pair of vertices $x$ and $y$ at distance $\ell$ in $G$, the number of vertices closer to $x$ than to $y$ is equal to the number of vertices closer to $y$ than to $x$. A complete characterization of $\ell$-distance-balanced corona products is given and a characterization of lexicographic products for $\ell \ge 3$, thus complementing known results for $\ell\in \{1,2\}$ and correcting an earlier related assertion. A sufficient condition on $H$ which guarantees that $K_n \,\square\, H$ is $\ell$-distance-balanced is given and it is proved that if $K_n \,\square\, H$ is $\ell$-distance-balanced, then $H$ is an $\ell$-distance-balanced graph. A known characterization of $1$-distance-balanced graphs is extended to $\ell$-distance-balanced graphs, again correcting an earlier claimed assertion.