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A prime harmonious labeling of a graph \(\varGamma \) with p nodes and q lines is a way of assigning distinct numbers from the set \(\{1,2,3,\dots ,2q-1\}\) to the vertices of the graph. For every edge that connects two vertices, we define a corresponding edge value in two steps: first, the two end vertices assigned numbers must be relatively prime, meaning their greatest common divisor is 1. Second, the value of the edge is obtained by adding the numbers of its two end vertices and then reducing the result modulo q. If these conditions are satisfied for all lines, then the labeling is called a prime harmonious labeling. Throughout the paper, we consider prime harmonious labeling as PH labeling and prime harmonious graph as PH graph. This paper explores Bull graph, Helm graph, Lobster graph, Double comb graph, \(K_{1,n}^{[1]} \circ K_{1,n}^{[2]}\) , \(S(k_{1,n})\) , H-graph, and Caterpillar graph. Next, we explore the applications of communication networks using PH labeling like Channel Assignment and Frequency Allocation, Secure Communication and Key Distribution, Routing and Data Integrity.

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Innovations in Prime Harmonious Labeling and Their Relevance to Communication Networks

  • 会议论文

  • 发表日期：

    2026年

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  • M. Bhuvaneswari,
  • R. Manoharan

摘要

A prime harmonious labeling of a graph \(\varGamma \) with p nodes and q lines is a way of assigning distinct numbers from the set \(\{1,2,3,\dots ,2q-1\}\) to the vertices of the graph. For every edge that connects two vertices, we define a corresponding edge value in two steps: first, the two end vertices assigned numbers must be relatively prime, meaning their greatest common divisor is 1. Second, the value of the edge is obtained by adding the numbers of its two end vertices and then reducing the result modulo q. If these conditions are satisfied for all lines, then the labeling is called a prime harmonious labeling. Throughout the paper, we consider prime harmonious labeling as PH labeling and prime harmonious graph as PH graph. This paper explores Bull graph, Helm graph, Lobster graph, Double comb graph, \(K_{1,n}^{[1]} \circ K_{1,n}^{[2]}\) , \(S(k_{1,n})\) , H-graph, and Caterpillar graph. Next, we explore the applications of communication networks using PH labeling like Channel Assignment and Frequency Allocation, Secure Communication and Key Distribution, Routing and Data Integrity.