Split Matching Polynomial Of Graphs Theory, Properties, And Applications

Abstract

The concept of graph polynomials has played a pivotal role in combinatorial graph theory and chemical graph theory. In this paper, we introduce a novel polynomial invariant called the Split Matching Polynomial (SMP), which extends the classical matching polynomial by incorporating structural constraints arising from split partitions of graphs. A split-valid matching is defined based on vertex partitions into cliques and independent sets, reflecting real-world structures such as molecular networks, biological systems, and communication frameworks. We rigorously define the SMP, prove its well-definedness, and derive recurrence relations and closed-form expressions for fundamental graph families including paths, cycles, and stars. The theoretical development is further extended to graph operations such as corona, join, and lexicographic products. Strong theorems are presented with complete proofs, exploring the algebraic and combinatorial properties of SMP. In the context of chemical graph theory, we show how SMP encodes meaningful insights about molecular stability and resonance energy, particularly in systems like benzenoid hydrocarbons, dendrimers, and nanostar networks. This study opens new avenues for exploring graph-based molecular descriptors and suggests several promising directions for future work including SMP-based topological indices and hybrid polynomials combining domination and matching characteristics.