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More on Zagreb coindices of graphs

Hua Hongbo (Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huai’an, Jiangsu , P.R. China)
Ashrafi Ali Reza (Department of Mathematics, Faculty of Science, University of Kashan, Kashan, I.R. Iran + School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, I.R. Iran)
Zhang Libing (Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huai’an, Jiangsu, P.R. China)

For a nontrivial graph G, its first and second Zagreb coindices are defined, respectively, as M1(G)= ∑uvE(G)(dG (u)+ dG (v)) and M2(G) = ∑uvE dG (u)dG(v), where dG (x) is the degree of vertex x in G. In this paper, we explore further properties of Zagreb coindices. First, we investigate Zagreb coindices of two classes of composite graphs, namely, Mycielski graph and edge corona, and we present explicit formulas for Zagreb coindices of these two composite graphs. Then we we give two estimations on Zagreb coindices of graphs in terms of the number of pendent vertices and Merrifield-Simmons index, respectively. Finally, we give several Nordhaus-Gaddum type bounds for the first Zagreb coindex.

Keywords: Zagreb coindices, composite graphs, Nordhaus-Gaddum type bounds, the number of pendent vertices