Edge Quasi \(\lambda\)-distance-balanced Graphs in Metric Space

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.