(untitled) (European Journal of Pure and Applied Mathematics, 2025)
Graphs considered in this paper are finite simple undirected graphs. Let G = (V(G), E(G)) be a graph with the vertex set V(G) = {x_1,x_2,..., x_n}, for some positive integer n. The vertex space V(G) of G, is a vector space over the field Z_2 = {0,1}. The elements of V(G) are all the subsets of V(G). Vector addition is defined as A + B = A \:\: B, the symmetric difference of sets A and B, for all A,B V(G). Scalar multiplication is defined as 1 · A = A and 0 · A = , for all A V(G). The subgraph G S of G induced by a subset S of V(G), is the largest subgraph whose vertex set is S. The vertex-uniform set V_H(G) of a subgraph H with respect to G, is the set of all elements of V(G) that induces a subgraph isomorphic to H. The span of V_H(G) shall be denoted by V_H(G). If V_H(G) is a generating set, that is V_H(G) = V(G), then H is called a vertex-generator subgraph of G. This study determines some vertex-generator subgraphs of complete bipartite graph K_m,n and tadpole graph T_n,m.