Study On The Structural Parameters Based On Eccentricity Of Graphs

In this thesis,we only consider simple,undirected and connected graphs.For a connected graph G,with degG(vi)and ?G(vi)denoting the degree and eccentricity of the vertex vj,the non-self-centrality number and the total irregularity of G are defined as N(G)=?|?G(vi)-?G(vi)| and irrt(G)=? |degG(vj)-degG(vi)|,respectively,with summations over all pairs of vertices.In this paper,we focus on relations between these two structural invariants.It is proved that irrt(G)>N(G)holds for almost all graphs.Some graphs are constructed for which irrt(G)=N(G).Moreover,we prove that irrt(T)<N(T)for any tree T of order n?15 with diameter d?2n/3 and maximum degree 3.The first and second Zagreb eccentricity indices of graph G are defined as:(?)and(?),respectively,where ?G(vi)denotes the eccentricity of vertex vi in G.The eccentric complexity Cec(G)of G is thenumber of different eccentricities of vertices in G.In this paper we present some results on thecomparison between E1(G)/n and E2(G)/m for any connected graphs G of order n with medges,including general graphs and the graphs with given Cec For convenience,we define M property,E property,and L property of the graph,respectively.We prove that G has M or E property when G is a graph with diameter d?2.G has E property if and only if G is a self-centered graph.Simultaneously,for the Cartesian product G and H of the graphs G and H,it is proved that GH has the L property,if both G and H have L property with avd(H)+avd(G)?c(H)/(H)+?c(G)/?(G),in this formula,(?),(?).Moreover,We prove that,for Cec(G)=2,there are different graphs thathave the M property,E property and L property,respectively.