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.
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