A graph geodesic polynomial is an algebraic expression linked with its polynomial that gathers information about geodesics, which means shortest paths in a graph connecting two vertices. Examining these polynomials’ characteristics and behaviors, and how they interact with the underlying network, graph structure is necessary for researching the geodesic polynomials of graphs and their vertices. In this paper, we present geodesic polynomials for fan graphs, merging of two star graphs, hericyclic graphs formed by attaching n edges at one or all vertices of C3, Helm graph and geodesic polynomial at a vertex for star graphs and the extended star graphs.