Non-self-centrality Number Of Graphs And Transversal Domination Number Of Some Graphs

In this thesis,we study the non-self-centrality number of graphs and the independent transversal domination number of grids and tubes.In the first part of this thesis,we study an eccentricity-based invariant of a graphs,which is a novel eccentricity-based invariant of a graph called the non-self-centrality number of graphs,which was introduced by X-u et.In[On a novel eccentricity-based invariant of a graph,Xu et.al.Acta Mathematica Sinica,English Series,32(12)1477-1493(2016)].The non-self-centrality number of graph G,denoted by N(G),is defined as(?)where ei is the eccentricityof vertex vi.In the above paper,among other results,the authors determine the lower and upper bounds on N(G)and characterize the extremal graphs achieving the bounds.However,the characterization given in there is not complete.Here we present the full list of the extremal graphs.Furthermore,we determine the lower bound on N(T)for trees T of order n with diameter d and the upper bound on N(G)for bicyclic graphs.Finally,as the consequence of our results,we obtain the upper bound on N(G)for tricyclic graphs G of odd order.In the second part,we study the independent transversal domination number in grid and tube graphs,i.e.,the cardinality of a smallest dominating set which intersects each maximum independent set,there are some known results on the independent transversal domination number of graphs.Based on these results,we continue study on this invariant of graphs.A set(?)V of vertices in a graph G =(V,E)is called a dominating set,if every vertex in V-S is adjacent to a vertex in S and minimum cardinality of a dominating set is called the domination number.A dominating set which intersect every maximum independent set in G is called an independent transversal dominating set.The mini-mum cardinality of an independent transversal dominating set is called the independent transversal dominating set of G.As our result,we determine the independent transversal domination number of some grid and tube graphs.