The Estrada index of evolving graphs

The Estrada index of a simple graph G with adjacency matrix A   is defined as EE(G)=trace(eA)EE(G)=trace(eA). It can be used in a range of fields as an efficient measuring tool. An evolving graph G1,G2,…,GNG1,G2,…,GN is a time-ordered sequence of graphs over a fixed set of vertices. By using the natural definition of a walk on the evolving graph that respects the arrow of time, we extend the (static) Estrada index to accommodate this promising dynamic setting. We find that although asymmetry is raised intrinsically by time’s arrow, the dynamic Estrada index is order invariant. We establish some lower and upper bounds for the dynamic Estrada index in terms of the numbers of vertices and edges. Illustrative examples are worked out to demonstrate the computations involved.