Abstract
This paper explores Subgroup Product Graphs (SPG) in cyclic groups, presenting a Vertex Degrees Formula based on the prime factorization of a positive integer n. The Isolated Vertex Property asserts that for a positive integer n, the SPG γ_sp(G) lacks isolated vertices. The Matrix Degree and Edge Formula provide a matrix representation and calculate the edges in SPG. Additionally, a Subgraph Relation identifies the complete graph Kπ(n) as a subgraph in γ_sp(G). Specific Examples illustrate vertex degrees for different n values. In essence, the study contributes isomorphisms, characterizes properties, and computes degrees and edges for diverse subgroups in Subgroup Product Graphs.
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