(untitled) (European Journal of Pure and Applied Mathematics, 2024)
Graphs considered in this paper are finite simple graphs, which has no loops and multiple edges. Let G= (V(G),E(G)) be a graph with E(G) = {e_1, e_2,…, e_m}, for some positive integer m. The edge space of G, denoted by E(G), is a vector space over the field _2. The elements of E(G) are all the subsets of E(G). Vector addition is defined as X+Y = X Δ Y, the symmetric difference of sets X and Y, for X,Y E(G). Scalar multiplication is defined as 1· X =X and 0 · X = for X E(G). Let H be a subgraph of G. The uniform set of H with respect to G, denoted by E_H(G), is the set of all elements of E(G) that induces a subgraph isomorphic to H. The subspace of E(G) generated by E_H(G) shall be denoted by E_H(G). If E_H(G) is a generating set, that is E_H(G)= E(G), then H is called a generator subgraph of G. This paper determines some generator subgraphs of the square of a cycle. Moreover, this paper established sufficient conditions for the generator subgraphs of the square of a cycle.