On the Resistance-Harary Index of Graphs Given Cut Edges

Graphs are often used to describe the structure of compounds and drugs. Each vertex in the graph represents themolecule and each edge represents the bond between the atoms.The resistance distance between any two vertices is equal to the resistance between the two points of an electrical network.TheResistance-Harary index is defined as the sum of reciprocals of resistance distances between all pairs of vertices. In this paper, the extremal graphs withmaximumResistance-Harary index are determined in connected graphs with given vertices and cut edges.


Introduction
Recently, the development of computational chemistry owes much to the theory of graphs.One of the most popular areas is topological index.The molecular topological index can describe the molecular structure quantitatively and analyze the structure and performance of molecules.
Among them, the most common topological index is resistance distance.The resistance distance is raised by Klein and Randić [1] as a distance function.Let  be a simple connected graph with vertex set () and edge set ().
The resistance distance between vertices  and V of  is recorded as   (, V), which represents the effective resistance between two nodes  and V in an electronic network; that is to say, the vertex corresponds to the node of the electronic network, and the edge corresponds to the unit resistance.
Similar to the traditional path distance, the resistance distance not only has good mathematical characteristics but also has good physical characteristics [2,3]; at the same time, it also has a good application in chemistry.
Harary index is another kind of graph invariants proposed by Plavšić et al. [4] and by Ivanciuc et al. [5] in 1993 for the characterization of molecular graphs.Name this in honor of Professor Frank Harary's birthday.The Harary index () is defined as the sum of reciprocals of distances between all pairs of vertices of the graph ; that is, Gutman [6] and Xu [7] investigated the Harary index of trees; they studied the Harary index of tree and pointed out that the path and the star attain the minimal and maximal value of Harary index, respectively, among a tree with given n vertices.In recent years, the Harary index was well studied in mathematical and chemical literatures [8].
The reciprocal resistance distance is also called electrical conductance, Klein and Ivanciuc [9] investigated QSAR and QSPR molecular descriptors computed from the resistance distance and electrical conductance matrices, and they proposed the global cyclicity index () as where the sum is over all edges of .
In [10], using graph theory, electronic networks, and real number analysis methods, Yang obtains some conclusions about the global cyclicity index.
Following the definition of the Harary index, Chen et al. [11] generalized the global cyclicity index and introduced Resistance-Harary index, defined as In [11], Chen et al. depicted the graphs with largest and smallest Resistance-Harary index in all unicyclic graphs.All graphs considered in this paper are finite and simple.Before proceeding, we introduce some further notation and terminology.A cut edge is an edge whose deletion increases the number of components.Denote by   ,   , and   the complete, cycle, and star graph on  vertices, respectively.For a graph  with V ∈ (),  − V denotes the graph resulting from  by deleting V and its incident edges.For an edge V of the graph  (the complement of , resp.),  − V ( + V, resp.)denotes the graph resulting from  by deleting (adding, resp.)V.
For other definitions, we can refer to [12].
In this thesis, we consider the Resistance-Harary index of graphs given cut edge.We will determine the graphs with maximum Resistance-Harary index in connected graphs given vertices and cut edges.

Some Preliminary Results
We first list or prove some lemmas as basic but necessary preliminaries.
Proof.Suppose that  is not a complete graph.Then there exists a pair of vertices V  and V  in  such that V  V  ∈ (). Let We only to prove that ) is similar.We distinguish the following two cases.
Case 1. V  and V  are vertices of cycle   in , where  is the length of   .
Let  1 (V  , V  ) and  2 (V  , V  ) be distance between V  and V  in the cycle   , respectively.By the definition of resistance, we have That is to say, Case 2. V  and V  are not vertices in any cycle of .
In this case, we have This completes the proof.
This completes the proof.Lemma 5. Let  and   be the graphs in Figure 1, where  0 is a complete graph, and then () ≤ (  ), with equality if and only if Proof.From the definition of the Resistance-Harary index and Lemma 4, we have Similarly, Since  0 is a complete graph,   0 (,  1 ) =   0 (,   ) = 2/ 0 for 1 ≤  ≤ .And with equality if and only if   0 (  ,   ) = 0; that is,  1 =  2 = ⋅ ⋅ ⋅ =   .This completes the proof.Lemma 6.Let  and   be the graphs depicted in Figure 2, where  2 and  3 are all complete graphs.Let   be the number of vertices of   , where  = 2, 3.If  3 >  2 and ( 4 ) may be an empty set, then RH(  ) > RH().
Proof.By the definition of the Resistance-Harary index and Lemmas 1 and 4, we have Since  2 and  3 are all complete graphs, for any two vertices ,  ∈ (  ), we have    (, ) = 2/  , where  = 2, 3.
Similarly, we have RH We just need to prove that In fact, In the following, we just need to prove that the fraction of ∇ 1 is greater than zero.Denote ∇ 2 to be the fraction of ∇ 1 ; in fact, Since  3 >  2 , RH(  ) > RH().This completes the proof.
Figure 3: The graphs in Lemma 7 and in Theorem 9.
Proof.By direct calculation, we have Similarly, we can deduce the value of RH(  ).