Edge Quasi $\lambda$-distance-balanced Graphs in Metric Space

TL;DR

This paper introduces and analyzes edge quasi-$\lambda$-distance-balanced graphs in metric spaces, exploring their properties, constructions, and extensions to related graph classes, with applications to graph indices and product operations.

Contribution

It defines and studies the properties of edge quasi-$\lambda$-distance-balanced graphs, including their bipartiteness, index calculations, and extensions to related graph classes.

Findings

Every $EQDBG$ is bipartite.

Calculated the edge-Szeged index for $EQDBG$.

Extended the concept to nicely and strongly edge distance-balanced graphs.

Abstract

In a graph $A$, the measure $|M_g^A(f)|=m_g^A(f)$ for each arbitrary edge $f=gh$ counts the edges in $A$ closer to $g$ than $h$. $A$ is termed an edge quasi-$\lambda$-distance-balanced graph in a metric space (abbreviated as $EQDBG$), where a rational number ($>1$) is assigned to each edge $f=gh$ such that $m_g^A(f)=\lambda^{\pm1}m_h^A(f)$. This paper introduces and discusses these graph concepts, providing essential examples and construction methods. The study examines how every $EQDBG$ is a bipartite graph and calculates the edge-Szeged index for such graphs. Additionally, it explores their properties in Cartesian and lexicographic products. Lastly, the concept is extended to nicely edge distance-balanced and strongly edge distance-balanced graphs revealing significant outcomes.