Wiener Index, Harary Index, and Szeged Index of the Total Graph on Z_(p^k )

Wahyu Ulyafandhie Misuki; Sugi Guritman; Bib Paruhum Silalahi; Sita Armi Aulia

doi:10.31764/jtam.v10i4.39846

Wiener Index, Harary Index, and Szeged Index of the Total Graph on Z_(p^k )

Wahyu Ulyafandhie Misuki, Sugi Guritman, Bib Paruhum Silalahi, Sita Armi Aulia

Abstract

Although total graphs over Z_(p^k ) have been investigated previously, the structural and topological consequences of defining adjacency using the set of nonzero zero divisors have not been explored. Moreover, the Szeged index for this class of total graphs has not yet been established. This study investigates the total graph over Z_(p^k ) with adjacency defined by nonzero zero divisors. Using algebraic and graph-theoretic methods, a coset-based partition induced by the canonical homomorphism onto Z_(p^k ) is employed to characterize the graph structure, determine its distance distribution, and derive its topological indices. The total graph is shown to be disconnected and decomposes into components determined by pairs of cosets, each of which is a modified complete or complete bipartite graph. Closed-form formulas for the Wiener, Harary, and Szeged indices are derived as functions of p and k, and each connected component is shown to have bounded diameter. These results provide a structural characterization and closed-form formulas for the Wiener, Harary, and Szeged indices of this class of total graphs, thereby extending algebraic graph theory and enriching the study of topological graph invariants.

Keywords

Wiener Index; Harary Index; Szeged Index Total Graph; Modulo Integer Ring.

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DOI: https://doi.org/10.31764/jtam.v10i4.39846