Average Eccentricity Of Direct Product Of Graphs And Average Connectivity Of Mycielskian Of A Graph

The study of graph index is a very important part of graph theory. These indices havea very wide range of applications in computer science, combinatorial chemistry, physicsand other applied sciences. We hope to use all possible information to describe thevarious feature of graphs and make it apply in further research and practical application.There are several types of graph indices, especially those based on vertex distances andconnectivity, such as, Wiener index W, Hyper-Wiener index W W, P I index, Hosoyaindex, ABC index, Wiener polarity index Wp, Szeged index Sz, eccentric connectivityindex etc.For a graph G of order n, the average eccentricity ecc(G) of G is defined as ecc(G)=1nu∈V (G)ε(u), where ε(u) is the eccentricity of the vertex u in G. The average connec-tivity k(G) is defined as the average of the connectivities between all pairs of vertices ofG, that is, k(G)=1∑(n, v), where kG(u, v) is the connectivity of vertices2)u,v∈V (G)kG(uu and v in G. In this paper, we establish a lower bound on the average eccentricity ofdirect product of graphs and calculate this parameter for benzenoid parallelogram Bm,n.Moreover, we present a lower bound on the average connectivity of Mycielskian of a graph.