On the inverse-order divisor graph of the cyclic group of order 2p

Onjefu Amali; Abolape Deborah Akwu; Paul Augustine Ejegwa; Johnson Aderemi Awolola; Suleiman Hamza Isah

doi:10.46481/jnsps.2027.3750

Authors

Onjefu Amali amalionjefu@gmail.com Department of Mathematics, Joseph Sarwuan Tarka University, Makurdi, Benue, Nigeria
Abolape Deborah Akwu Department of Mathematics, Joseph Sarwuan Tarka University, Makurdi, Benue, Nigeria
Paul Augustine Ejegwa Department of Mathematics, Federal University of Health Sciences, Otukpo, Benue, Nigeria
Johnson Aderemi Awolola Department of Mathematics, Joseph Sarwuan Tarka University, Makurdi, Benue, Nigeria
Suleiman Hamza Isah Department of Mathematics, Nasarawa State University, Keffi, Nasarawa, Nigeria

Keywords:

Inverse-order divisor graph, Inverse graph, Order-divisor graph, Hybrid graph, Cyclic group

Abstract

This paper introduces the inverse-order divisor graph associated with a finite group G, a hybrid graph obtained by combining the adjacency conditions of the order-divisor graph and the classical inverse graph. Our primary focus is on the additive cyclic group of order 2p, where p denotes an odd prime. By partitioning the vertices according to the element orders, we obtain an explicit characterization of the adjacency relations of this hybrid graph. Using this characterization, we derive exact formulas for the size of the graph and the degree of each vertex. We further establish that the resulting graph is connected without being complete. In addition, we determine several fundamental metric and combinatorial properties, including the domination number, radius, diameter, girth, clique number, and chromatic number. These results establish several theoretical properties of the inverse-order divisor graph associated with Z_2p for every odd prime p and contribute to the broader study of algebraic hybrid graphs.

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