On Average Distance of Neighborhood Graphs and Its Applications

Abstract

Graph invariants such as distance have a wide application in life, in particular when networks represent scenarios in form of either a bipartite or non-bipartite graph. Average distance μ of a graph G is one of the well-studied graph invariants. The graph invariants are often used in studying efficiency and stability of networks. However, the concept of average distance in a neighborhood graph G′ and its application has been less studied. In this chapter, we have studied properties of neighborhood graph and its invariants and deduced propositions and proofs to compare radius and average distance measures between G and G′. Our results show that if G is a connected bipartite graph and G′ its neighborhood, then radG1′≤radG and radG2′≤radG whenever G1′ and G2′ are components of G′. In addition, we showed that radG′≤radG for all r≥1 whenever G is a connected non-bipartite graph and G′ its neighborhood. Further, we also proved that if G is a connected graph and G′ its neighborhood, then and μG1′≤μG and μG2′≤μG whenever G1′ and G2′ are components of G′. In order to make our claims substantial and determine graphs for which the bounds are best possible, we performed some experiments in MATLAB software. Simulation results agree very well with the propositions and proofs. Finally, we have described how our results may be applied in socio-epidemiology and ecology and then concluded with other proposed further research questions.