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http://dx.doi.org/10.14317/jami.2015.365

THE ZAGREB INDICES OF BIPARTITE GRAPHS WITH MORE EDGES  

XU, KEXIANG (Department of Mathematics, College of Science, Nanjing University of Aeronautics & Astronautics)
TANG, KECHAO (Department of Mathematics, College of Science, Nanjing University of Aeronautics & Astronautics)
LIU, HONGSHUANG (Department of Mathematics, College of Science, Nanjing University of Aeronautics & Astronautics)
WANG, JINLAN (Department of Mathematics, College of Science, Nanjing University of Aeronautics & Astronautics)
Publication Information
Journal of applied mathematics & informatics / v.33, no.3_4, 2015 , pp. 365-377 More about this Journal
Abstract
For a (molecular) graph, the first and second Zagreb indices (M1 and M2) are two well-known topological indices, first introduced in 1972 by Gutman and Trinajstić. The first Zagreb index M1 is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M2 is equal to the sum of the products of the degrees of pairs of adjacent vertices. Let $K_{n_1,n_2}^{P}$ with n1 $\leq$ n2, n1 + n2 = n and p < n1 be the set of bipartite graphs obtained by deleting p edges from complete bipartite graph Kn1,n2. In this paper, we determine sharp upper and lower bounds on Zagreb indices of graphs from $K_{n_1,n_2}^{P}$ and characterize the corresponding extremal graphs at which the upper and lower bounds on Zagreb indices are attained. As a corollary, we determine the extremal graph from $K_{n_1,n_2}^{P}$ with respect to Zagreb coindices. Moreover a problem has been proposed on the first and second Zagreb indices.
Keywords
Vertex degree; Zagreb index; Bipartite graph;
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