Eccentric distance sum index for some classes of connected graphs

In this paper we show some properties of the eccentric distance sum index which is defined as follows ξ(G) = ∑ v∈V (G)D(v)ε(v). This index is widely used by chemists and biologists in their researches. We present a lower bound of this index for a new class of graphs.


Introduction.
In this paper we will be considering simple and connected graphs.We will start with a few definitions.Let G = (V (G), E(G)) be a simple connected graph of order n = |V (G)| and size m = |E(G)|.
For a vertex v ∈ V (G), we denote a set of neighbours of v by N (v).Degree is denoted by deg(v) and defined as deg(v) = |N (v)|.Vertex of degree equal to 1 is called a pendant vertex.For vertices u, v ∈ V (G), we define a distance d(u, v) as the length of the shortest path between u and v. What is more, D(v) denotes the sum of all distances from the vertex v.The eccentricity ε(v) of a vertex v is the maximum of distances between v and all other vertices.The minimum eccentricity over all vertices is denoted by rad(G) and called the radius of the graph G, while the maximum eccentricity is denoted by diam(G) and called the diameter of the graph G.
Let K n be a complete graph and P n be a path on n vertices.For graphs G and H we denote the join operation by G + H and by G ∪ H we mean a disjoint sum of those graphs.
A vertex in a graph is called a cutpoint when the number of components in a graph increases after removal of this vertex.Graph which is connected, nontrivial and has no cutpoints is called nonseparable graph.A block of a graph G is a maximal nonseparable subgraph of G.A cactus is a connected graph, each of whose block is isomorphic to a cycle or a path of order 2. A spanning tree of a connected graph G is a subtree of G which includes all of the vertices of G.Not defined notations one can find in [1].
The eccentric distance sum index is defined as follows: The eccentric distance sum index was introduced in 2002 by Gupta, Singh and Madana [2].The authors showed that this graph invariant can be used for predicting some biological and physical properties.It has a vast potential in quantitative structure-activity relationship.Some structureactivity studies using the eccentric distance sum index were proved [2] to be better than the corresponding values obtained using the Wiener index of a graph defined in 1947 [6] as: The eccentric distance sum index properties were studied recently.Yu, Feng and Ilić [7] described the extremal tree with respect to the eccentric distance sum index among all n-vertex trees.They proved it also for unicyclic graphs.Hua, Xu, Wen [4] gave then a short and unified proof of their results.
Ilić, Yu and Feng [5] showed also some lower and upper bounds for the eccentric distance sum index in terms of the Wiener index, degree distance and some other graph invariants.

Eccentric distance sum index for cacti and some other classes of connected graphs.
In this section there is a research done for lower bound of the eccentric distance sum index for some generalization of cacti.In Theorem 2.1 we present an interesting result of Hua, Xu and Wen [4] for cacti.
Theorem 2.1 (Hua, Xu and Wen [4]).Let G be a cactus with n ≥ 4 vertices and Proof.By the definition of the eccentric distance sum index: Let us now consider another graph structure than cactus.Let n, k 2 , k 3 be integers with k 2 , k 3 ≥ 0 and n ≥ 2k 2 + 3k 3 + 1.Let G n,k 2 ,k 3 be a class of connected graphs of order n consisting of blocks: k 2 cycles with no chords, k 3 cycles with one chord and paths P 2 .Some examples are presented in Figure 1 (note that we have For this class of graphs we present the lower bound of the eccentric distance sum index and it is an extended result of Theorem 2.1.The idea of the proof is based on the proof of Theorem 2.1.
Proof.We are considering a graph G from a class G n,k 2 ,k 3 .
Let S i be the set of vertices with eccentricity equal to i and n i = |S i |.Let us consider first n 1 > 0. Let v be a vertex with ε(v) = 1.Then each vertex u ∈ V (G)\{v} is adjacent to v.As n ≥ 5 and G ∈ G n,k 2 ,k 3 , then G can only be a graph obtained by introducing k 3 independent paths P 3 and k 2 independent paths P 2 among pendant vertices of a star K 1,n−1 .That is G ∼ = G n,k 2 ,k 3 .An example of this graph you can see in Figure 2. Since n 1 = 1, we have the following result: (for a vertex v with ε(v) = 1) Let us now consider the case where G ∈ G n,k 2 ,k 3 with n 1 = 0.Here we have ε(v) ≥ 2 for every vertex v in a graph.By Lemma 2.2 we have By the structure of the graph we have m = n − 1 + 2k 3 + k 2 , where n − 1 is the number of edges in a spanning tree of our graph and 2k 3 + k 2 is the sum of edges which do not belong to this spanning tree.
The result is as follows: This completes the proof.Theorem 2.3 cannot be expanded for n = 4 with k 3 = 1 since in this case Remark 2.4.After applying k 3 = 0 in Theorem 2.3, we immediately get the result of Theorem 2.1 for n ≥ 5.
In Theorem 2.5 we will give a lower bound for ξ d (G) for the class of graphs defined below.
Let p, q be positive integers, where q ≥ p ≥ 1 and let k p , k p+1 , . . ., k q be a sequence of integers, where k i ≥ 0 for p ≤ i ≤ q; k p , k q ≥ 1 and n = 1 + q i=p k i i.Let G be a class of connected graphs of order n with k i blocks isomorphic to K 1 + P i , p ≤ i ≤ q.Numbers p, q are the lengths of the shortest and the longest paths P i , respectively.This class is audited in the next theorem.Theorem 2.5.Let G ∈ G be a graph of order n ≥ 5. Then with the equality if and only if G ∼ = K 1 + q i=p k i P i , where n = 1 + q i=p k i i and p, q are the lengths of the shortest and the longest paths P i , respectively.
Proof.For our graph G the number of vertices with eccentricity equal to one (denoted by n 1 ) is n 1 ≤ 1 as n ≥ 5. Case 1.If ε(v) = 1 for a vertex v in G, then every vertex u ∈ V (G)\{v} is adjacent to v. Now we know that in this situation G can only be a graph isomorphic to K 1 + q i=p k i P i .Now we will show how to compute ξ d (K 1 + q i=p k i P i ).There is only one vertex v for which ε(v) = 1 and it is clear that D(v) = n − 1.For every other vertex u in G we have ε(u) ≥ 2. Each introduced path P i has two "ends" (every "end" of the path has exactly two vertices at the distance equal to one and it has the distance equal to two to every other vertex) and (i − 2) internal vertices of P i (every internal vertex has exactly three vertices at the distance equal to one).
Note that the number of pendant vertices is n − 1 − q i=p ik i .So, this is what we have: ik i (for vertices with degree 1) Case 2. Let us now consider the case when n 1 = 0.In this case we have ε(v) ≥ 2 for every vertex v in G.By Lemma 2.2 we have ξ d (G) ≥ 4n(n − 1) − 4m.
We also have m = n − 1 + q i=p k i (i − 1).Hence This is the end of the proof.

Conclusions.
In this paper we presented a lower bound for the eccentric distance sum index for some generalization of cacti.This result extends the result of Hua, Xu and Wen [4] for cacti.There remains an open problem of how to order graphs in a class by the values of the eccentric distance sum index.Note that G n,k 2 ,k 3 cannot be ordered by ξ d (G) for n = 7, k 2 = 1, k 3 = 1.
In the future we will study the problem mentioned above for n > 7.

Lemma 2 . 2 .
with the equality if and only if G ∼ = Cat n,k 2 , where Cat n,k 2 is the cactus obtained by introducing k 2 independent edges among pendant vertices of n-vertex star K 1,n−1 .Let G be a graph of order n and size m.If rad(G) ≥ 2, then ξ d (G) ≥ 4n(n − 1) − 4m with equality holding if and only if rad(G) = 2.