Abstract
Let G be a connected (molecule) graph. The Wiener index W G and Kirchhoff index K f G of G are defined as the sum of distances and the resistance distances between all unordered pairs of vertices in G , respectively. In this paper, explicit formulae for the expected values of the Wiener and Kirchhoff indices of random pentachains are derived by the difference equation and recursive method. Based on these formulae, we then make comparisons between the expected values of the Wiener index and the Kirchhoff index in random pentachains and present the average values of the Wiener and Kirchhoff indices with respect to the set of all random pentachains with n pentagons.
Highlights
All graphs considered in this paper are simple, undirected, and connected
All notations not defined in this paper can be found in [1]
Let G be a graph with the vertex set V(G) v1, v2, . . . , vn and edge set E(G). e traditional distance d(vr, vs) between vertices vr and vs is the number of edges of a shortest path connecting these vertices in G
Summary
All graphs considered in this paper are simple, undirected, and connected. All notations not defined in this paper can be found in [1]. Is is the oldest topological index related to molecular branching [6] It is defined as the sum of distances between all unordered pairs of vertices in G, i.e., W(G). Where r(vi|G) denotes the sum of resistance distances between vi and all other vertices of G, defined by n r vi|G rvi, vj. The Kirchhoff index based on resistance distance is a graph invariant and molecular structure descriptor. Motivated by the works in [42,43,44], in this paper, we derive explicit formulae for the expected values of the Kirchhoff indices of random alpha-pentachain [43] and spiro-pentachain as shown in Figure 1 and Wiener index of random spiropentachain. Denote by E[Ξ] the expected value of a random variable
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