Research Article | Open Access
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: ; ; , where W, H denote the Wiener 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.
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 .
We consider simple graphs. Let be a connected graph with the vertex-set . 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 and in . The reciprocal distance matrix of is also called the Harary matrix .
For , denotes the set of its neighbors in . Let and be, respectively, the maximum eigenvalues of and ; the distance spectral radius of is the largest -eigenvalue . Ivanciuc et al.  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 C6–C10 alkanes.
Recall the Hosoya definition of the Wiener index  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 , 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 half-sum of the off-diagonal elements of the reciprocal molecular distance matrix :
The reciprocal distance matrix can be simply obtained by replacing all off-diagonal elements of the distance matrix by their reciprocals: It should be noted that diagonal elements are all equal to zero by definition. This matrix was first mentioned by Balaban et al. . The maximum eigenvalues of various matrices have recently attracted attention of mathematical chemists [6–8].
The Harary index and the related indices such as its extension to heterosystems  and the hyper-Harary index  have shown a modest success in structure-property correlations , 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.
2. An Interesting Phenomenon
Recall that, for a positive integer and set , the circulant graph is the graph with vertices, labeled with integers modulo , such that each vertex is adjacent to other vertices . The set is called a symbol of . As we will consider only undirected graphs without loops, we assume that and if and only if , and therefore the vertex is adjacent to vertices 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 ).
Now, consider 4-circulant graphs with 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 and .
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 1 (see ). Let be an nonnegative irreducible symmetric matrix with row sums . If is the maximum eigenvalue of , then with either equality if and only if or there is a permutation matrix such that where all the row sums of are equal.
Lemma 2 (see ). Let be a r-regular graph of order . If are the adjacency eigenvalues of , then its D-eigenvalues are and , where D-eigenvalues denote the eigenvalues of the distance matrix .
Lemma 3 (see ). Let be a r-circulant graph on vertices and the Wiener index of ; then the following equality holds: where denotes the number of edges.
Lemma 4 (see ). Let be a component of and . If , where is adjacent to in , let be obtained by removing an edge of and add a new edge . For all , if a vertex satisfies , then , where denotes the distance between a pair of vertices in .
3. Main Results
In this section, we will prove the following two theorems.
Theorem 5. Let be a r-circulant graph on vertices; then the following equality holds:
Proof. Obviously, and are irreducible for . By Lemma 1, we have Note that is a regular graph of degree ; hence all the row sums of and are equal, respectively; that is, , ; then we have 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, . By (13) and (15), we have
On the other hand, Indulal proved the following fact in .
Fact. Let be a graph with Wiener index . Then and the equality holds if and only if is distance regular.
Then, from (16) and the previous fact, we have .
By equality (11), we obtain . This completes the proof of the theorem.
4. 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 ) with equality on the left if and only if and equality on the right if and only if , where is the path and 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 .
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 . Then we show the following theorem.
Theorem 7. Let be a simple connected graph on vertices and cut edges; then .
Proof. We will use the above well known fact to prove , with equality if and only if .
Let be set of cut edges; assume any block of has a clique. If , then , and the proposition holds.
Now suppose that ; let denote the components of , respectively, where .
Claim 1 (, ). In fact, if , such that , let , and . Let . Obviously, is a disconnected graph on vertices and cut edges, and . Due to in by Lemma 4, we have , and then we reach a contradiction.
Without loss of generality, assume .
Claim 2. If is adjacent to , , then or .
In fact, if and .
Let denote two components containing in , respectively, and .
Let . Then still is a connected graph with vertices and cut edges, and , where is a pendent edge in . Using Lemma 4, we have , and this leads to a contradiction.
Since is connected, according to Claims 1 and 2, we obtain . Hence we have .
This completes the proof of the theorem.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The author is grateful to the reviewers for their valuable comments and suggestions. Project is supported by Hunan Provincial Natural Science Foundation of China no. 13JJ3118.
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