The Wiener Index of Circulant Graphs

Circulant graphs are an important class of interconnection networks in parallel and distributed computing. In this paper, we discuss the relation of the Wiener index and the Harary index of circulant graphs and the largest eigenvalues of distance matrix and reciprocal distance matrix of circulants. We obtain the following consequence: W/λ = H/μ; 2W/n = λ; 2H/n = μ, where W, H denote theWiener index and the Harary index and λ, μ denote the largest eigenvalues of distance matrix and reciprocal distance matrix of circulant graphs, respectively. Moreover we also discuss the Wiener index of nonregular graphs with cut edges.


Introduction
A circulant graph is a graph whose adjacency matrix (with respect to a suitable vertex indexing) can be constructed from its first row by a process of continued rotation of entries.The interest of circulant graphs in graph theory and applications has grown during the last two decades; they appeared in coding theory, VLSI design, Ramsey theory, and other areas.Recently there is vast research on the interconnection schemes based on circulant topology.Circulant graphs represent an important class of interconnection networks in parallel and distributed computing [1].
We consider simple graphs.Let  be a connected graph with the vertex-set () = {V 1 , V 2 , . . ., V  }.The distance matrix  of  is an  ×  matrix (  ) such that   is just the distance (i.e., the number of edges of a shortest path) between the vertices V  and V  in .The reciprocal distance matrix D   of  is also called the Harary matrix [2].
For V  ∈ (),   (V  ) denotes the set of its neighbors in .Let () and () be, respectively, the maximum eigenvalues of D and D  ; the distance spectral radius of  is the largest -eigenvalue ().Ivanciuc et al. [3] proposed to use the maximum eigenvalues of distance-based matrices as structural descriptors; they have shown that () and () are able to produce fair QSPR models for the boiling points, molar heat capacities, vaporization enthalpies, refractive indices, and densities for C 6 -C 10 alkanes.
Recall the Hosoya definition of the Wiener index [4] and the Harary index: Since the distance matrix and related matrices, based on graph-theoretical distance, are rich sources of many graph invariants (topological indices) that have been found to be used in structure-property-activity modeling [5], it is of interest to study spectra and polynomials of these matrices.
The Harary index,  = (), of a molecular graph  with  vertices is based on the concept of reciprocal distance and is defined, in parallel with the Wiener index, as the halfsum of the off-diagonal elements of the reciprocal molecular distance matrix D   = D  (): The reciprocal distance matrix D   can be simply obtained by replacing all off-diagonal elements of the distance matrix D  by their reciprocals: It should be noted that diagonal elements (D  )  are all equal to zero by definition.This matrix was first mentioned by Balaban et al. [6].The maximum eigenvalues of various matrices have recently attracted attention of mathematical chemists [6][7][8].
The Harary index and the related indices such as its extension to heterosystems [7] and the hyper-Harary index [8] have shown a modest success in structure-property correlations [9], but the use of these indices in combination with other descriptors appears to be very efficacious in improving the QSPR (quantitative structure-property relationship) models.
The paper is organized as follows.In Section 2 we bring forward an interesting phenomenon.In Section 3, we analyze the cause of this phenomenon to emerge; we report our results for the maximum eigenvalues of the Wiener matrix and the Harary matrix of -circulant graphs.Finally, in Section 4, we discuss the Wiener index of nonregular graphs.

An Interesting Phenomenon
Recall that, for a positive integer  and set  ⊆ {0, 1, 2, . . .,  − 1}, the circulant graph (, ) is the graph with  vertices, labeled with integers modulo , such that each vertex  is adjacent to  other vertices { +  (mod ) |  ∈ }.The set  is called a symbol of (, ).As we will consider only undirected graphs without loops, we assume that 0 ∉  and  ∈  if and only if  −  ∈ , and therefore the vertex  is adjacent to vertices  ±  (mod ) for each  ∈ .In other words, a graph is circulant if it is Cayley graph on the circulant group; that is, its adjacency matrix is circulant.
For any circulant graph , it is a regular graph.A circulant graph is called regular of degree (or valency) , when every vertex has precisely  neighbors.Let  denote the diameter of ; then  is less than the number of distinct eigenvalues of the adjacency matrix of  (see [10]).Now, consider 4-circulant graphs with 6, 7, . . ., 18 vertices, respectively.By a straightforward mathematical calculation, we obtain some data on the Wiener index: , the Harary index: , and the eigenvalues: ,  of the Wiener matrix and the Harary matrix as follows in Table 1.
From Table 1, it is easy to observe that / and / are equal.Moreover, we have 2/ =  and 2/ = .
We again observe Table 2.
From Table 2, we also obtain the same results previously when the number of vertices is fixed in (, ).
Why does this phenomenon occur?We will be interested in trying to explain something about this phenomenon.To prove our result, we need a few more lemmas.
Lemma 3 (see [13]).Let  be a r-circulant graph on  vertices and the Wiener index  of ; then the following equality holds: where || denotes the number of edges.

Main Results
In this section, we will prove the following two theorems.
Theorem 5. Let  be a r-circulant graph on  ( ≥ 4) vertices; then the following equality holds: Note that  is a regular graph of degree ; hence all the row sums of D and D  are equal, respectively; that is, On the other hand, according to definitions of the Wiener index and the Harary index, we have Thus we obtain Theorem 6.Let  be a r-circulant graph on  vertices; then its Wiener index  and Harary index  listed below have the following relationship: Proof.Since a -circulant graph is a -regular graph, applying Lemma 2, we get its maximum distance matrix eigenvalue: For -regular graphs  = (, ) on  vertices the following equality holds: and we denote by || the number of edges of .According to ( 6) and ( 14) we have That is, 2 ≥ 2( − 1) − .By ( 13) and ( 15), we have On the other hand, Indulal proved the following fact in [15].
Fact.Let  be a graph with Wiener index .Then  ≥ 2/ and the equality holds if and only if  is distance regular.

Nonregular Graphs
As we just said, for a regular, those conclusions above hold.But for nonregular graph, it is not easy to tell whether or not the conclusions also hold.
For arbitrary tree on  vertices, It is proved that (see [16]) with equality on the left if and only if   ≅  It is well known that the Wiener index of  has a direct proportion with the distance spectral radius of  as vertices fixed; that is, the distance spectral radius of  is strictly increasing with the Wiener index increasing [15].Then we show the following theorem.Theorem 7. Let  be a simple connected graph on  ( ≥ 4) vertices and  cut edges; then () ≥ (   ).
Since  is connected, according to Claims 1 and 2, we obtain  ≅    .Hence we have () ≥ (   ).This completes the proof of the theorem.

Table 1
with either equality if and only if  1 =  2 = ⋅ ⋅ ⋅ =   or there is a permutation matrix Q such that

Table 2
1,−1 and equality on the right if and only if   ≅   , where   is the path and  1,−1 is the star on  vertices.Without loss of generality, we now discuss the Wiener index of graph    ; it is obtained by joining  pendent edges to a vertex of complete graph  − .