Connective Steiner 3-eccentricity index and network similarity measure

https://doi.org/10.1016/j.amc.2020.125446Get rights and content

Highlights

  • The connective Steiner 3-eccntricity index of graphs based on the 3-eccnetricity was introduced firstly. Some mathematical properties of connective Steiner 3-eccntricity index were present.

  • An O(n2)-polynomial time algorithm to compute the connective Steiner 3-eccntricity index of trees was developed.

  • Network similarity measure based on the connective Steiner 3-eccntricity index was built. By numerical analysis, we analyzed its advantage.

Abstract

For a set SV(G) in a network G, the Steiner distance dG(S) of S is the minimum size among all connected subnetworks whose vertex sets contain S. The Steiner k-eccentricity ɛk(v) of a vertex v of G is the maximum Steiner distance among all k-vertex set S which contains the vertex v, i.e., εk(v)=max{d(S)|SV(G),|S|=k,vS}. Based on Steiner k-eccentricity, the connective Steiner k-eccentricity index is introduced. As a newly structural invariant, some properties of the connective Steiner 3-eccentricity index are investigated. Firstly we present an O(n2)-polynomial time algorithm to calculate the connective Steiner 3-eccentricity index of trees. Secondly some optimal problems among some network classes are discussed. As its application, finally we consider the network similarity measure based on the connective Steiner 3-eccentricity index. By two different methods, we study its advantages. Numerical results show that the measure based on the connective Steiner 3-eccentricity index has more advantages than the ones based on other topological indices (graph energy, Randić index, the largest adjacent eigenvalue, the largest Laplacian eigenvalue).

Introduction

The canonical network form is a (undirected or directed) graph. Let us begin with the formal definitions and notations of graphs that represent networks. Throughout this paper, all graphs we considered are simple, undirected and connected.

Let G=(V(G),E(G)) be a simple, connected and undirected graph with vertex set V(G) and edge set E(G). The distance dG(u, v) between two vertices u, v ∈ V(G) in G is the length of a shortest path connecting u and v. The eccentricity ɛ(v) of a vertex v is the maximal distance between v and any other vertex, i.e., ε(v)=max{dG(u,v)|uV(G)}. Note that the distance also equals to the minimum size of a connected subgraph containing these two vertices. Based on this, the Steiner distance was introduced by Chartrand, Oellermann, Tian and Zou [4] in 1989 as a generalization of general distance. For a set SV(G) of k (k ≥ 2) vertices, a Steiner tree of S is the minimum connected subgraph of G which contains all vertices of S. The Steiner distance dG(S) (for short, d(S)) among the vertices S (or simply the Steiner distance of S) is the minimum size among all connected subgraphs whose vertex sets contain S. Note that if H is a connected subgraph of G such that SV(H) and |E(H)|=d(S), then H is a Steiner tree of S. So d(S)=min{|E(T)|SV(T)} where T is a Steiner tree of S. The classical distance is a Steiner 2-distance. The Steiner k-eccentricity ɛk(v) of a vertex v of G is defined by εk(v)=max{d(S)|SV(G),|S|=k,vS}. It is evident that the classical eccentricity is a Steiner 2-eccentricity.

As molecular descriptors, some distance-based graph invariants play an important role in chemical graph theory and are used to establishing correlations of chemical structures with various physical properties, chemical reactivity, or biological activity (see [1], [2], [8], [9], [11], [12], [13], [15], [19], [20], [21], [22], [23]). The connective eccentricity index was introduced to investigate the antihypertensive activity of derivatives of N-benzylimidazole [14]. Furthermore, some mathematical properties of connective eccentricity index were investigated [25], [26], [27]. For a graph G, the connective eccentricity index ξce(G) of G is defined byξce(G)=vV(G)d(v)ε(v).

Here we generalize it to Steiner k-eccentricity as follows. The connective Steiner k-eccentricity index ξkce(G) of a graph G is defined byξkce(G)=vV(G)d(v)εk(v).Alternatively, the connective Steiner k-eccentricity index can be written asξkce(G)=vivjE(G)(1εk(vi)+1εk(vj)).(1)

In this paper we study the connective Steiner 3-eccentricity index as a starting point. This paper is organized as follows. In Section 2, an O(n2)-polynomial time algorithm to calculate the connective Steiner 3-eccentricity index of trees is present. In Section 3, we optimize the extremal values of the connective Steiner 3-eccentricity index among some graph classes with given some graph parameters such as independence number, matching number, vertex connectivity, minimum degree and the number of pendent vertices. Finally we determine the maximal connective Steiner 3-eccentricity index among the cacti on n vertices and obtain the optimal graph. In Section 4, we employ the connective Steiner 3-eccentricity index to study the network similarity measure. Numerical results show that the measure based on the connective Steiner 3-eccentricity index has more advantages than the ones based on other topological indices (graph energy, Randić index, the largest adjacent eigenvalue, the largest Laplacian eigenvalue).

Section snippets

An algorithm of connective Steiner 3-eccentricity index of trees

The problem of finding the Steiner distance of a set of vertices is called the Steiner Problem and is NP-complete [10]. It seems that the problem of finding the Steiner k-eccentricity of a vertex is NP-complete. But to our best knowledge, it is still unsolved. In this paper we present an algorithm to calculate the connective Steiner 3-eccentricity index of trees based on the expression (1).

We calculate the connective Steiner 3-eccentricity index on a tree by the Algorithm 1. Mainly Algorithm 1

Upper best optimization of connective Steiner 3-eccentricity index

As mentioned in the above section, the computational complexity has been not determined. So optimizing the values of some network classes is natural and interesting. In this section we shall optimize the values of the connective Steiner 3-eccentricity index among some network classes with given some structural parameters. Let G be a set of graphs. If G is the unique graph which attains the maximal value c of connective Steiner 3-eccentricity index among all graphs in G, then G is called upper

Network similarity measure

Similarity measure between distinct networks plays fundamental roles in the field of network analysis, which has been an interesting and useful task in some disciplines such as applied mathematics, information science and so on. Up to now exploring methods to measure the similarity between distinct networks has been a current research item as various class of networks exist, moreover, not every measure is generally applicable. Different measure methods may be used to different networks. In this

Conclusion

In this paper we introduce the connective Steiner k-eccentricity index which is a newly graph invariant based on Steiner k-eccentricity. Some mathematical properties of this variant are given. We consider its computational complexity for trees. An O(n2)-polynomial time algorithm to calculate the connective Steiner 3-eccentricity index of trees is present. Moreover, we consider the optimal problems among some network classes. As its application, we employ it to explore the network similarity

Funding

This work is supported by Guizhou Talent Development Project in Science and Technology (KY[2018]048), Foundation of Guizhou University of Finance and Economics (2019XJC04).

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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