The Zagreb Indices And Modification Indices Of Graphs

All graphs considered in this thesis are undirected and simple. For a (molecular) graph, the firstand second Zagreb indices () are two well-known topological indices in chemical graph theoryintroduced in1972by Gutman and Trinajsti. In this thesis, we studied the Zagreb indices and theirmodified variants (including multiplicative versions of Zagreb indices, multiplicative Zagrebcoindices) of graphs and obtained sharp upper and lower bounds on the these indices in some specificsets of graphs. Moreover, the corresponding extremal graphs were characterized at which the upperand lower bounds on these indices were attained. The first chapter we introduced the basic concepts ofgraph theory and related research background. And in the second chapter we determined the sharpupper and lower bounds on the Zagreb indices of graphs from (the set of bipartite graphs obtainedby deleting edges from complete bipartite graph,where) and characterized the correspondingextremal graphs at which the upper and lower bounds on Zagreb indices were attained. Furthermore,we determined the extremal graphs from with respect to Zagreb coindices. In the third chapter westudied the Narumi-Katayama index, multiplicative versions of Zagreb indices, such as multiplicativeZagreb index and multiplicative sum Zagreb index. In it we showed that, in G(n, k), graph had themaximal Narumi-Katayama index, the maximal multiplicative Zagreb index and maximalmultiplicative sum Zagreb index where denoted G(n,k) the set of graphs of order n and with chromaticnumber k. And the extremal graphs from G(n, k) with k=2or3were determined with minimalvalues of these above indices. In the fourth chapter we defined the multiplicative Zagreb coindices ofgraphs, and obtained some properties of them. In the fifth chapter we listed some related openproblems to this topic.