Extremal Polyphenyl Chains Concerning κ-matchings And κ-independent Sets And Vertex PI Indices Of Some Sums Of Graphs

Denote by A_n the set of the polyphenyl chains with n hexagons. For any A_n∈A_n,let mk(A_n) and ik(A_n) be the numbers of k-matchings and k-independent sets of A_n, re-spectively. In chapter 1 of this paper, we show that for any polyphenyl chain A_n∈A_n andfor any k≥0, m_k(M_n)≤m_k(A_n)≤m_k(O_n) and i_k(M_n)≥i_k(A_n)≥i_k(On), with leftequalities holding for all k only if A_n = M_n, and the right equalities holding for all k onlyif A_n = O_n, where M_n and O_n are a meta-chain and a ortho-chain, respectively. We alsoobtain their matching and independence related properties.The vertex Padmakar-Ivan (PI_v) index of a graph G is defined as the sum of [meu(e|G)+mev(e|G)] over all edges e = uv of a connected graph G, where meu(e|G) is the number ofvertices of G lying closer to u than to v, and mev(e|G) is the number of vertices of G lyingcloser to v than to u. In chapter 2 of this paper, we give the explicit expression of the vertexPI indices of some sums of graphs.Theorem 2.2.2. For any A_n∈A_n, n≥3,m_k(M_n)≤m_k(A_n)≤m_k(O_n).Moreover, the above equality holds for each k if and only if A_n = M_n or A_n = O_n.Theorem 2.3.2. For any A_n∈A_n and each k≥0,i_k(O_n)≤i_k(A_n)≤i_k(M_n)Moreover, the above equalities hold for each k if and only if A_n = O_n or A_n = M_n.Theorem 3.2.1. Let G₁ and G₂ be two connected graphs. ThenPI_v(G₁ +S G₂) = (|V₁| + |E₁|)(|V₁||V₂||E₂| - |V₁|n(G₂) + 2|V₂|²|E₁|)Let e = uv be an edge of a graph G. We denote by N(u,v)(G) the set of all verticesu of G satisfying d(u,u ) = d(v,u ), and by n(u,v)(G) the cardinality of N(u,v)(G) and letn(G) =∑uv∈E(G)n(u,v)(G).Theorem 3.2.2. Let G₁ and G₂ be two connected graphs. ThenPI_v(G₁ +R G₂) = (|V₁| + |E₁|)[(|V₁||V₂||E₂|-|V₁|n(G₎ + 3|V₂|²|E₁|)] - |V₂|²n(R(G₁))...