Some Extremal Trees and Unicyclic Graphs w.r.t. Non-self-centrality Number

Abstract

The concept of centrality of a graph is introduced by Camille Jorden in 19th century for the analysis of different network models which is widely used in facility location problems. To measure the non-self-centrality extent of a graph Xu et al. introduced an eccentricity based graph invariant called Non-self-centrality number (NSC number). The centrality concept and eccentricity measures of a graph is used in network sciences, opimization theory, facility location problem, chemical graph theory and many more. In this thesis we have considered some problems of extremal graph theory with respect to this distance based graph invariant NSC number. We have considered the class T(n; p) of all non-self centered tree graphs of order n with p pendant vertices. We found out the unique maximal graph Dn;p with respect to NSC number among all the graphs in T(n; p) and also formulated the mathematical expressions for it and hence gave the upper bound. Further we extended our study to find the maximal graph among a class of unicyclic graphs Un(3; ) with some fixed parameters, that is, fixed degree and atmost three central vertices. We found the unique graph eUn;3 which attains the maximum value of NSC number in this class. v