A Linear Algebraic Framework for Geodesic Distribution Vectors of Connected Graphs

For a connected graph \(G\), let \(f_i(G)\) denote the number of geodesics of length \(i\). The vector \(F(G)=\left(f_1(G), f_2(G), \ldots, f_{n-1}(G)\right)\) is called the geodesic distribution vector of \(G\), where \(n\) is the order of the graph. The collection of all geodesic distribution vectors of connected graphs on \(n\) vertices generates a finite-dimensional real vector space, called the geodesic distribution space. In this paper, we introduce a canonical family of connected graphs \(G_{n, i}\) obtained by identifying one end vertex of a path with a vertex of a complete graph. We classify all geodesics of these graphs and derive explicit formulas for their geodesic distribution vectors. It is shown that these vectors form a basis having a triangular coordinate structure, from which we establish that the dimension of the space is \(n-1\). Finally, we characterize the dual space and prove that every linear functional on the geodesic distribution space is uniquely determined by the geodesic frequencies. These results provide an algebraic framework for the study of geodesic distributions and establish a direct connection between shortest-path structures and finite-dimensional vector spaces.