  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.
