Asian Journal of Mathematics and Computer Research

In this study, we present a balanced graph (L_Δ (G)), which is an edge-based graph transformation that improves classical linear graphs by incorporating the vertex degree as a structural node in (L_Δ (G)). The two edges are adjacent only if they are not adjacent and have a length distance of 2 and are connected via an intermediate vertex w of deg(w)≥3.Through this formula, adjacencies mediated by vertices with reduced degrees are excluded, making \(\bar{L(G}\)) is a sub graph. It is clear to us from the definition that (L_Δ (G)), is continuous if and only if δ(G) ≥ 3, which reflects on the adjacency property, which is very weak, giving it a special uniqueness not found in traditional linear Apleyan graphs. We also define lower limits for the dominance number: γ(L_Δ (G)) ≥ \(\frac{|E(G)|}{2(Δ(G)−1)^2+1}\)

An upper bound on the matching number: μ(L_Δ (G)) ≤|E(G)|- α(L_Δ (G)) and for the independence number α(L_Δ (G))≥ μ(G)

The results obtained for these limits are accurate for third-order regular graphs, confirming that (L_Δ (G)), embodies the correlation between degree heterogeneity and edge independence. This transformation provides a solid foundation for the design of global networks, communication systems, and performance, and addresses the fundamental shortcomings of L_Δ (G)and (\(\bar{L(G}\))) linear graph models, which fail to accommodate the structural heterogeneity based on the degree of the vertex. It provides the first solid foundation for modeling structurally constrained systems where local degree centrality is a critical factor making it uniquely suitable for designing nested networks.