Further properties on the degree distance of graphs

Abstract

In this paper, we study the degree distance of a connected graph $$G$$G, defined as $$D^{'} (G)=\sum _{u\in V(G)} d_{G} (u)D_{G} (u)$$D?(G)=?u?V(G)dG(u)DG(u), where $$D_{G} (u)$$DG(u) is the sum of distances between the vertex $$u$$u and all other vertices in $$G$$G and $$d_{G} (u)$$dG(u) denotes the degree of vertex $$u$$u in $$G$$G. Our main purpose is to investigate some properties of degree distance. We first investigate degree distance of tensor product$$G\times K_{m_0,m_1,\cdots ,m_{r-1}}$$G×Km0,m1,?,mr-1, where $$K_{m_0,m_1,\cdots ,m_{r-1}}$$Km0,m1,?,mr-1 is the complete multipartite graph with partite sets of sizes $$m_0,m_1,\cdots ,m_{r-1}$$m0,m1,?,mr-1, and we present explicit formulas for degree distance of the product graph. In addition, we give some Nordhaus---Gaddum type bounds for degree distance. Finally, we compare the degree distance and eccentric distance sum for some graph families.