Further properties on the degree distance of graphs

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 ) d G ( u ) D G ( u ) , where $$D_{G} (u)$$ D G ( u ) is the sum of distances between the vertex $$u$$ u and all other vertices in $$G$$ G and $$d_{G} (u)$$ d G ( 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 × K m 0 , m 1 , ⋯ , m r - 1 , where $$K_{m_0,m_1,\cdots ,m_{r-1}}$$ K m 0 , m 1 , ⋯ , m r - 1 is the complete multipartite graph with partite sets of sizes $$m_0,m_1,\cdots ,m_{r-1}$$ m 0 , m 1 , ⋯ , m r - 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.