Generalized Euler Function Graphs
Abstract
Let and be positive integers and . For a positive divisor of , we define the generalized Euler function graph of type to be the graph whose vertex set is the set of integers in where the greatest common divisor of and is , and two vertices and are adjacent if and only if the greatest common divisor of and is . The generalized Euler function graph of type is the Euler function graph which has already been studied. In this research, we focus on studying the generalized Euler function graphs of types and . We explore properties of vertices, degree and connectivity of the graphs. Moreover, we present relationships among those graphs, relatively prime graphs and Eulerian graphs. Keywords : greatest common divisor ; relatively prime graph ; divisor graphReferences
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West, D.B. (2001). Introduction to Graph Theory. New Jersey: Prentice Hall.
Wilson, R.J. (1985). Introduction to Graph Theory. New York: Longman.
Garcia, P.G., & Ligh, S. (1983). A generalization of Euler's ϕ-Function. Fibonacci Quart., 21, 26-28.
Koshy, T. (2007). Elementary Number Theory with Applications. (2nd edition). Boston: Mass.
Madhavi, L. (2002). Studies on Domination Parameters and Enumeration of Cycles in Some Arithmetic Graphs.
Ph.D. Thesis. Tirupati: S.V. University.
Manjuri, M., & Maheswari, B. (2012). Matching dominating sets of Euler-Totient-Cayley graphs. IJCER., 2, 103-107.
Manjuri, M., & Maheswari, B. (2013). Clique dominating sets of Euler totient Cayley graphs. IOSR-JM., 4, 46-49.
Pomerance, C. (1983). On the longest simple path in the divisor graph. Congr. Numer., 40, 291-304.
Sangeetha, K.J., & Maheswari, B. (2015). Edge domination in Euler-Totient-Cayley graph. IJSER., 3, 14-17.
Shanmugavelan, S. (2017). The Euler function graph G(ϕ(n)). Int. J. Pure Appl. Math., 116, 45-48.
West, D.B. (2001). Introduction to Graph Theory. New Jersey: Prentice Hall.
Wilson, R.J. (1985). Introduction to Graph Theory. New York: Longman.
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Published
2023-01-04
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Research Article
