Algebraic Structures of Lexicographic Products of Open Neighborhood Graphs

Rudrapati Bhuvaneswara Prasad, Etigowni Mahojani

Abstract: This paper develops a systematic algebraic framework for studying lexicographic product graphs of open neighborhood graphs. We introduce and investigate ring-theoretic structures, module-theoretic properties, ideal generation, and algebraic independence conditions arising from such graph products. Extending the edge-parameter results of earlier combinatorial studies, we establish isomorphism theorems between quotient ring structures and graph-derived algebras, characterize the Jacobson radical of the associated graph rings, and prove spectral conditions linked to the regularity of the underlying graphs. For cycle graphs Cn (n ≥ 4), complete graphs Kn (n ≥ 3), and complete bipartite graphs Kn,n (n ≥ 2), we derive explicit algebraic invariants including annihilator ideals, projective dimensions, and homological dimensions of associated modules. All results are supported by algebraic verification using symbolic computation. The non-commutativity and non-associativity of the lexicographic product are shown to produce non-commutative ring structures with rich ideal lattices, offering new tools for applications in coding theory, cryptographic graph algebras, and network topology.

Keywords: lexicographic product, open neighborhood graph, graph ring, algebraic independence, commutative algebra

How to Cite?: Rudrapati Bhuvaneswara Prasad, Etigowni Mahojani, "Algebraic Structures of Lexicographic Products of Open Neighborhood Graphs", Volume 15 Issue 5, May 2026, International Journal of Science and Research (IJSR), Pages: 307-312, https://www.ijsr.net/getabstract.php?paperid=SR26505102319, DOI: https://dx.doi.org/10.21275/SR26505102319

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