On the Adjacent Eccentric Distance Sum Index of Graphs.

For a given graph G, ε(v) and deg(v) denote the eccentricity and the degree of the vertex v in G, respectively. The adjacent eccentric distance sum index of a graph G is defined as [Formula in text], where [Formula in text] is the sum of all distances from the vertex v. In this paper we derive some bounds for the adjacent eccentric distance sum index in terms of some graph parameters, such as independence number, covering number, vertex connectivity, chromatic number, diameter and some other graph topological indices.

Introduction

In theoretical chemistry and biology, molecular structure descriptors or topological indices are used for modeling information of molecules, including physico-chemical, toxicologic, pharmacologic, biological and other properties of chemical compounds. A topological index is a numerical value based on a simple graph, which is an invariant. Topological indices have been found to be useful in chemistry, bioinformatics and network science. Generally, these indices are divided into three classes. One class is degree-base indices [51], such as (general) Randic index [46, 52], (general) zeroth order Randic index [34, 35], Zagreb index [26], ABC index [50]. One class is distance-based indices [61], such as Wiener index [40, 57, 58, 60], Balaban index [4, 5, 16], the Wiener polarity index [24, 49], Szeged index [1], Harary index [2], Hosoya index [32, 33]. The third class is spectrum-based indices, such as graph energy [18, 27, 31, 36, 37, 47, 48], incidence energy [9, 10], matching energy [12, 13, 43], Randic energy [8, 19], HOMO-LUMO index [45], Kirchhoff index [44], graph entropies [11, 14, 15, 20, 21, 42, 65]. Recently, some other indices based on eccentricity have been introduced and investigated by mathematicians, chemists and biologists, which contain eccentric connectivity index [28, 41, 55], eccentric distance sum [29, 38, 62], augmented and super augmented eccentric connectivity indices [3, 25], connective eccentricity index [30, 63, 64], adjacent eccentric distance sum index [53, 54] and so on.

Throughout this paper, graphs considered are simple and connected. Let G be a simple connected graph with vertex set V(G). For a vertex v ∈ V(G), deg(v) denotes the degree of v. For vertices u, v ∈ V(G), the distance d(u, v) is defined as the length of the shortest path between u and v in G and D (v) (or D(v) for short) denotes the sum of all distances from v. The eccentricity ɛ(v) of a vertex v is the maximum distance from v to any other vertex. The radius r(G) of a graph is the minimum eccentricity of any vertex, while the diameter D(G) of a graph is the maximum eccentricity of any vertex in the graph. Let K and P be the complete graph and path graph on n vertices, respectively. We write K − ke for the graph obtained from K by deleting k independent edges.

In chemistry and biology, people want to find some new molecules with desired physicochemical properties, for example, boiling points, molecular volumes, flavor threshold concentration antiviral activity, energy levels, electronic populations, etc. And these properties can be quantitatively represented by some values of a certain index. The notion of a “topological index” appears first in a paper of the Japanese chemist Hosoya [32], who investigated the surprising relation between the physicochemical properties of a molecule and the number of its independent edge subsets (matchings). A topological index is a map from the set of chemical graphs to the set of real numbers. Therefore a topological index is a numeric quantity that is mathematically derived in a direct and unambiguous manner form the structural graph of a molecule. Since isomorphic graphs possess identical values for any given topological index, these indices are referred to as graph invariants. Topological indices usually reflect both molecular size and shape. The advantage of topological indices is that they may be used directly as simple numerical descriptors in a comparison with physical, chemical, or biological parameters of molecules in quantitative structure-property relationships (QSPR) and in quantitative structure-activity relationships (QSAR). Therefore, one of the most interesting problems on topological indices is to study the extremal properties of topological indices and characterize the extremal structures. Another direction is to study the relation between topological indices and some other graph invariant, which would help people to know the indices better and find more internal information.

The quantification of chemical structures is a basic problem in structure-activity relationships (SAR) which is an important tool for developing safer and potent drugs. The procedure for quantification of chemical structures is by translation of chemical structures into characteristic numerical descriptors (topological indices) [6, 7, 39, 56]. This procedure provides a correlation between quantitative biological activity and qualitative chemical structures. Based on this, Sardana and Madan [53] introduced a novel topological descriptor–adjacent eccentric distance sum index, which is defined as To verify that this index has a vast potential for SAR, Sardana and Madan [53, 54] selected different data sets to investigate some properties of adjacent eccentric distance sum index. In [53], to predict anti-HIV activity of 4,5,6,7-tetrahydro-imidazo-[4, 5, 1−jk][1, 4] benzodiazepin-2 (1H)-one (TIBO) derivatives, the authors introduced this index which is computable. Moreover, the authors compared property of the adjacent eccentric distance sum index with that of the eccentric connectivity index and derived some excellent correlations between anti-HIV activity and both the topological descriptors. Their results show that the adjacent eccentric distance sum index has a potential for structure-activity/property studies. As we know, Wiener index is one of the most studied topological indices from the view of theory and applications, see for details [22, 23, 59]. In [54], the authors investigated the relationship of Wiener index and adjacent eccentric distance sum index with the antioxidant activity of nitroxides, as well as their hydroxylamine and amine precursors. In their investigation, authors selected 68 analogues whose antioxidant activity was reported. Firstly they computed the values of the Wiener index and the adjacent eccentric distance sum index for each analogues. Subsequently, each analogue was assigned a biological activity which was then compared with the reported antioxidant activity as H 2 O 2 protection at a concentration of 100 μM. They supposed that compounds with an H 2 O 2 protection of 2.5 or more were active and those with H 2 O 2 protection of less than 2.5 or more were inactive. The overall degree of prediction was derived from the ratio of the total number of compounds predicted correctly to that of the total number of all compounds in both active and inactive ranges. Their results show that a total of 58 out of 68 compounds were classified correctly in both the active and inactive ranges with regard to Wiener index, but for adjacent eccentric distance sum index a total of 59 out of 68 compounds were classified correctly. The overall degree of prediction was found to 0.85 in case of Wiener index and 0.87 in case of adjacent eccentric distance sum index. These results encourage us to study the mathematical properties of this novel topological descriptor.

In this paper we derive some bounds for adjacent eccentric distance sum index in terms of some graph parameters, such as independence number, covering number, chromatic number, diameter and some graph topological indices.

Methods

Since adding edges would increase the degrees of some vertices and not increase ɛ(v), D(v) for any vertex v ∈ V(G), we have the following result.

Lemma 1 Let G be a non-complete connected graph and e be an edge in (the complement of G). Then ξ (G) > ξ (G+e).

The following result is immediate from Lemma 1.

Theorem 1 Let G be a connected graph on n vertices. Then with equality holding if and only if G ≅ K .

If G ≠ K , then with equality holding if and only if G ≅ K −e.

A graph is vertex-transitive if, for any two vertices u and v, there is an automorphism f of G such that f(u) = f(v). For a vertex-transitive graph G, it is regular and r(G) = ɛ(w) for any w ∈ V(G). The Wiener index of a graph G, denoted by W(G), is defined as the sum of the distances between all pairs of vertices in graph G, that is

Theorem 2 Let G be a vertex-transitive graph on n vertices with degree δ. Then

The total eccentricity of the graph G, denoted by ζ(G), is defined as the sum of eccentricities of all vertices of graph G, i.e., It is sometimes interesting to consider it [17].

Theorem 3 Let G be a connected graph on n ≥ 3 vertices. Then with equality holding if and only if G ≅ K .

Proof. By fact that D(v) ≥ deg(v) with equality holding if and only if ɛ(v) = 1 and deg(v) = n−1, i.e., G ≅ K , we have

An independent set of G is a subset S of V(G) such that no two vertices in S are adjacent in G. The independence number of G, denoted by α(G), is the size of a maximum independent set of G. Let 𝓖(n, α) be the set of all connected graphs with n vertices and independence number α. Let G ⋄ denote the graph . Let V(G ⋄) = V 1∪V 2, where V 1 = {v 1, v 2, ⋯, v } is the independent set of G ⋄ and V 2 = {v , ⋯, v } is the vertex set of K . Let e 1 be an edge joining two vertices in V 1 and V 2. Let e 2 be an edge incident to two vertices in V 2.

Theorem 4 Let G be a connected graph with n vertices and independence number α. Let be an integer. Then

with equality holding if and only if G ≅ G ⋄;

if G ∈ 𝓖∖{G ⋄}, then with equality holding if and only if G ≅ G ⋄−e 1 when 2 ≤ α ≤ C; G ≅ G ⋄−e 2 when α ≥ C+1.

Proof. Assume that G 0 is a graph with the minimal adjacent eccentric distance sum index among all connected graphs on n vertices with independence number α. Let S be a maximum independent set of G 0. Since adding edges would decrease the adjacent eccentric distance sum index, it follows that any vertex in V(G 0)−S is adjacent to every vertex in S and G 0−S is complete. So G 0 ≅ G ⋄. We have .

By deleting an edge in G ⋄, we get two graphs up to isomorphism: G ⋄−e 1, G ⋄−e 2. Since deleting an edge would increase the adjacent eccentric distance sum index, we only compare ξ (G ⋄−e 1) with ξ (G ⋄−e 2).

By some elementary calculations, we have So we have

Let be a function. Taking derivation for f(x), we have So f(x) is a strict decreasing function. It is easy to verify that f(x) ≥ 0 if and f(x) ≤ 0 if , which implies the result.

A covering of a graph G is a subset K of V(G) such that every edge of G has at least one end-vertex in K. The covering number of a graph G is the number of vertices in any minimal covering. Since the complement of the maximum independent set is just the minimum covering, we have the following result.

Theorem 5 Let G be a connected graph on n vertices with covering number γ. Then with equality holding if and only if .

A vertex cut of G is a subset V′ of V such that G−V′ is disconnected. A κ-vertex cut is a vertex cut of κ elements. The vertex connectivity of a graph G, denoted by κ(G), is the minimum κ for which G has a κ-vertex cut.

Theorem 6 Let G be a connected graph with n vertices and vertex connectivity κ. Then with equality holding if and only if G ≅ K ∨(K 1∪K ).

Proof. Let G 0 be a graph with the minimal adjacent eccentric distance sum index among all connected graphs on n vertices with vertex connectivity κ. Then there exists a vertex cut S ⊂ V(G 0) with |S| = κ such that G 0−S = G 1∪G 2∪⋯∪G , where G 1, G 2, ⋯, G (t ≥ 2) are connected components of G 0−S. Since adding edges will decrease the adjacent eccentric distance sum index, the following three claims hold: (1) t = 2; (2) the subgraphs G 1, G 2 and the induced subgraph G 0[S] are complete; (3) any vertex of G 1∪G 2 is adjacent to any vertex in S. Let |G | = n for i = 1,2 and n 1 ≥ n 2. Then G 0 ≅ K ∨(K ∪K ) and n 1+n 2 = n−κ.

By the definition of adjacent eccentric distance sum index, we have Let . Then we consider the difference

Therefore, G ≅ K ∨(K 1∪K ) has the minimal adjacent eccentric distance sum index among all connected graphs on n vertices with vertex connectivity κ.

Theorem 7 Let G be a connected graph with n vertices and minimum degree δ. Then with equality holding if and only if G ≅ K ∨(K 1∪K ).

Proof. Let G be a graph with the minimum adjacent eccentric distance sum index among connected graphs of order n with minimum degree δ. Assume that G has vertex connectivity κ.

Let be a function. It can be checked that and therefore f(x) is strictly decreasing. Hence f(κ) ≥ f(δ) due to the fact that κ ≤ δ.

By Theorem 6, we have ξ (G) ≥ f(κ) ≥ f(δ). If ξ (G) = f(δ), then κ = δ. This implies the result.

Similarly, we have

Theorem 8 Let G be a connected graph on n vertices with edge connectivity λ. Then with equality holding if and only if G ≅ K ∨(K 1∪K ).

Theorem 9 Let G be a connected graph on n vertices with maximum degree △. Then with equality holding if and only if G ≅ K △∨(K 1∪K ).

The chromatic number of a graph G, denoted by χ(G), is the minimum number of colors such that G can be colored with these colors and no two adjacent vertices have the same color.

Theorem 10 Let G a connected graph on n vertices with chromatic number χ. Assume that n = χs + r with 1 ≤ r < χ. Then with equality holding if and only if G ≅ T .

Proof. Let G 0 be a χ-chromatic graph with minimal adjacent eccentric distance sum index. G 0 must be of the form . For any vertex , ɛ(v) = 2, D(v) = n+n −2 and deg(v) = n−n . So we have Assume that n 1 ≤ n 2⋯ ≤ n . Let if 2 ≤ n < n .

Considering the difference Therefore, T has the minimal adjacent eccentric distance sum index among all connected graphs on n vertices with chromatic number χ. By simple calculations, we have

A diametrical path in a graph is a path whose length equals the diameter of the graph. Let P = v 0 v 1…v be a diametrical path in G. For the vertex v 0, the distance layer L is the set of vertices v ∈ G satisfying d(v 0, v) = i for i = 0, 1, …, D, i. e. In particular, is called the central layer.

The distance layer L is called trivial if |L | = 1. Obviously, L 0 is a trivial layer.

For , |l 1−l 2| ≤ 1 and l 1+l 2 = D−2, the graph is obtained from (the join of K and ) by identifying one end-vertex of each path of length l 1 and l 2 with u and v, respectively. It is evident that its noncentral layers are trivial and any subgraph induced by the union of the central layer and its neighboring layer is complete.

For odd D, denote by the set of graphs obtained from K ⋁K by connecting an end-vertex of a path P 1 = P ( with all vertices from K and connecting an end-vertex of a path P 2 = P ( with all vertices from K .

Theorem 11 Let G be a connected graph of order n with diameter D. Then with equality holding if and only if for even D and for odd D.

Proof. Let G 0 be a graph with the minimal adjacent eccentric distance sum index among all connected graphs on n vertices with diameter D. Assume that P = v 0 v 1⋯v is a diametrical path in G 0. Since adding an edge decreases the adjacent eccentric distance sum index, the subgraphs induced by the union of two neighboring distance layers in G 0 are complete.

Suppose that is the smallest number such that |V | > 1 in G 0. Let v be a vertex in V , but different to v . We construct a new graph from G 0 by removing v from V to V such that has the diameter D and the subgraphs induced by the union of two neighboring distance layers in are complete. By the definition of adjacent eccentric distance sum index, we have It follows that Note that for any 1 ≤ i ≤ k−1, we have ɛ(u) = ɛ(w) for u ∈ V , w ∈ V and For i = 0, we have

It follows that This contradict to the fact that G 0 has the minimal adjacent eccentric distance sum index in 𝓖.

Suppose that is the largest number such that |V | > 1. Let w be a vertex in V , but different to v . We construct a new graph from G 0 by removing w from V to V such that has the diameter D and the subgraphs induced by the union of two neighboring distance layers in are complete. As above, we can get that , a contradiction.

Therefore, is the unique graph with minimal adjacent eccentric distance sum index in 𝓖 for even D; for odd D, it can be seen that all graphs in have the same value and there are exactly extremal graphs with minimal adjacent eccentric distance sum index.

At last we shall calculate the value . Assume that P = v 0 v 1⋯v be a diametrical path in . We divide the vertex set into three parts: , and the central layer, i.e., .

When D is even, we have So we have

For v ∈ V 1(i ≥ 1), we have: ɛ(v ) = D−i,

, deg(v ) = 2; ɛ(v 0) = D,

, deg(v 0) = 1

For v ∈ V 2, we have: ɛ(v ) = i,

, deg(v ) = 2; ɛ(v ) = D,

, deg(v ) = 1

For v ∈ V 3, we have: , , deg(v) = n−D+1.

Similarly, if D is odd, for any , one has

Discussion

In this paper, we investigated the adjacent eccentric distance sum index and gave some lower bounds for this index in terms of some graph parameters such as independence number, covering number, vertex connectivity, chromatic number, diameter and some other topological indices. Moreover, the extremal graphs (attaining these bounds) are also characterized. Throughout this paper, the proof techniques used the structural properties of the graph parameters and the topological indices.

We only considered several graph parameters and topological indices in this work. To complete this knowledge, we shall consider relations between the adjacent eccentric distance sum index and other graph parameters and topological indices. In the future, we will continue to study extremal values of adjacent eccentric distance sum index for some special classes of graphs as well as for more general graphs. One direction of the future work is to consider the extremal values among some chemical or biology networks, such as benzenoid systems, fullerene graphs and polymeric networks.