Asian Journal of Pure and Applied Mathematics

The square graph Γ(G) of a finite group G is an undirected graph with vertex set G, in which two distinct vertices a and b are adjacent if and only if a² = b or b² = a. For a finite group G and a prime p, the p-power graph Γp (G) has vertex set G, with two distinct vertices a and b adjacent if and only if ap = b or bp = a. This paper examines square graphs and p-power graphs of finite cyclic groups. For square graphs, conditions determining the maximum degree are established. In particular, the maximum degree is 2 for cyclic groups of odd order greater than 3 and for the cyclic group of order 6, while it is 3 for cyclic groups of even order other than 2 and 6. The connected components of the square graph of a cyclic group of odd order are shown to be cycles, K2, or isolated vertices. Vertices of degree 3 are also characterised for cyclic groups of even order other than 6. For a positive integer n and a prime p, the maximum degree of Γp (\(\mathbb{Z}\)n) satisfies Δ(Γp (\(\mathbb{Z}\)n)) ≤ p+1. Moreover, when gcd(p, n) = 1 and n > p2, the maximum degree is 2. These results describe degree behaviour and connected-component structure within the graph constructions considered.