some graph parameters of indu-bala product of graphs

The indu-bala product of graphs g and h consists of two disjoint copies of the join of g and h such that there is an  adjacency between the corresponding vertices in the two copies of h. a vertex subset s of a graph g = (v, e) is said to be a geodetic set if every vertex in g is in some u−v geodesic, where u and v are any two vertices in s. the minimum cardinality of such a set is the geodetic number of g. the vertex subset d of a graph g is said to be a dominating set if every vertex in g is either in d or adjacent to at least one vertex in d. the minimum cardinality of such a set is the domination number of g. in this work, the authors studied various geodetic and dominating extensions with respect to the indu-bala product of graphs. the aα matrix associated with a graph is a convex linear combination of its adjacency matrix and degree diagonal matrix, offering deeper insights into the properties of both matrices. in this article the authors discuss the aα spectrum of indu-bala product of graphs.