Research On Several Classes Of Topological Indices Of Graphs And Related Combinatorical Structures

The thesis aims at discussing the calculational and extremal problems on the topological indices of graphs. It consists of six chapters.In chapter1, we gave a comprehensive survey of the backgrounds of topological indices of graphs.In chapter2, we introduced some elementary concepts of the topological indices, which will be used in this thesis.In chapter3, we discussed the extrmnal problems based on Schultz index and modified Schultz index. This chaper is consisted of five sections.In the first section, we introduced five types of grafting transformations, and discussed the changes on Schultz index and modified Schultz index after the transformations.In the second section, we determined unicyclic graphs with the maximum,minimum, the second maximum,the second minimum Schultz index and modified Schultz index. We got the following four theorems.Theorem1Let G be an arbitrary unicyclic graph with n vertices. Then S(G)≥3n2-3n-6, S*G)≥2n2+n-9, the equalities hold if and only if G(?)U(n-3,0,…,0).Theorem2Let G be an arbitrary unicyclic graph with n vertices, and G(?)U(n-3,0,…,0). Then S(G)≥3n2+n-22, S(G)≥2n2+5n-22, the equalities hold if and only if G(?)U(n-4,1,0,…,0).Theorem3Let G be an unicyclic arbitrary graph with n vertices. Then S(G)≤2/3n3-20/3n+14, S*(G)≤2/3n3-29/3n+23, the equalities hold if and only if G=L(n,3).Theorem4Let G be an arbitrary unicyclic graph with n vertices, and G(?)(n,3). Then S(G)≤2/3n3-32/3n+24, S*(G)≤2/3n3-41/3n+31,the equalities hold if and Only if G(?)L(n,3,n-5).In the third section,we determined bicyclic graphs with the minimum,the second minimum.the third minimum Schultz indeX and modified Schultz index.We obtained the following three theorems.Theorem5Let G be an arbitrary bicyclic graph with,7veI-tices. Then S(G)≥3n2+n-18, S*(G)≥2n2+7n-19, the equalities hold if and only if G(?)θn1(3,3).Theorem6Let G be an arbitrary bicyclic graph with n vertices, and G(?)θn1(3,3). Then S(G)≥3n2+n-16, S'(G)≥2n2+13n-13,the equalities hold if and only if G(?)Sn(3,3).Theorem7Let G be an arbitrary bicyclic graph with n vertices, and G(?)θn1(3,3),G(?)Sn(3,3).Then S(G)≥3n2+5n-42,S*(G)≥2n2+11n-39, the equalities hold if and only if G(?)G8.In the fourth section,we determined tricyclic graphs with the minimum,the second minimum Schultz index and modified Schultz index.We obtained the following two theorems.Theorem8Let G be an arbitrary tricyclic graph with n vertices. then(1)S(G)≥3n2+5n-32,the equality hold if and Only if G(?)R2,2,2,2or G(?)I3,3,3,2n-5；(2)S*(G)≥2n2+13n2-30,the equality holds if and only if G(?)R2,2,2,2,2,2n-4.Theorem9Let G be an arbitrary tricyclic graph with n vertices, and G(?)R2,2,2,2,2,2n-4,then(1)S(G)≥3n2+5n-28,the equality holds if and only if G(?)H2,3,3,3n-6；(2)S*(G)≥2n2+13n-27,the equality holds if and only if G(?I3,3,3,2n-5.In the fifth section,we determined cactuses with the minimum,the second minimum Schultz index and modified Schultz index.We arrived at the following two theorems.Theorem10Let G∈C(n.r)(1≤r≤|n-1/2|,then S(G)≥4rn-10r+3n2-7n+4S*(G)≥4r2+6rn-16r+2n2-5n+3, the equalities hold if and only if G(?)G0(n,r).Theorem11Let G∈C(n,r),and G(?)G0(n,r),thenS(G)≥r(4n-8)+3n2-3n-14,S*(G)≥4r2+6r(n-2)+2n2-n-17, the equalities hold if and only if G(?)G0(n,r).In Chapter4,we investigated the Merrifield-Simmons index of multi-bridge graphs.We got the following two theorems.Theorem12Letθ(a1,a2,…,ak)∈Θnk,a1≥1,then i(θ(a1,a2,…,ak))≤2k-1F(n+1-k)+F(n+2-k), the equality holds if and only ifθ(a1,a2,…,ak)(?)θ(1,1,…,1,n-k-1).Theorem13Letθ(a1,a2,…,ak)∈Θnk,k≥5,then i(θ(a1,a2,…,ak))≥3k-2(n-2k+4)+2k-1F(n-2k+3), the equality holds if and only if θ(a1,a2,…,ak)(?)θ(0,2,2,…2,n-2k+2).In Chapter5.we calculated the Schultz index of nanotubes covered by C4C8, and the geometric-arithmetic of nanotubes TUC4[p,q] TUVC6[2p,q],TUHC6[2p,Q],TUC4C8(S)[p,q], TUC4C8(R)[p,q].In Chapter6,we summarized our results and pointed out the further research ideas.