Bounds on the number of closed walks in a graph and its applications
© Chen and Qian; licensee Springer. 2014
Received: 29 September 2013
Accepted: 6 May 2014
Published: 20 May 2014
Using graph-theoretical techniques, we establish an inequality regarding the number of walks and closed walks in a graph. This inequality yields several upper bounds for the number of closed walks in a graph in terms of the number of vertices, number of edges, maximum degree, degree sequence, and the Zagreb indices of the graph. As applications, we also present some new upper bounds on the Estrada index for general graphs, bipartite graphs, trees and planar graphs, some of which improve the known results obtained by using the algebraic techniques.
MSC:05C50, 94C15, 05C38.
A walk W of length k starting at a vertex and ending at a vertex in G is a sequence of vertices, i.e., , in which is adjacent to for each . In particular, if the vertices (except the possible and ) are pairwise distinct, then W is well known as a path, and if then W is called a closed walk. It is well known  that the number of closed walks of length k in G is exactly the trace of which, in turn, is the sum of the k th power of the eigenvalues of G (known as the k th spectral moment of G). This fact is of importance in the theory of total π-electron energy, for details see [3, 4] and the references cited therein. Also, the sequence of the numbers of closed walks of length k, , starting at a given vertex v, was proposed by Randić  for characterization of the environment of vertex v.
where are the eigenvalues of G. The Estrada index has successfully found applications in various fields, including biochemistry [6, 7] and complex networks . Also, a number of mathematical properties, especially various lower and upper bounds on the Estrada index of a graph have been established, for details we refer the reader to [9–16]; other properties can be found in [17–19] and a latest survey paper by Gutman et al. .
In general, counting the closed walks in a graph (of large order) is not an easy work. Only a few results were obtained for some special types of graphs, e.g., vertex-transitive graphs  and generalized de Bruijn graphs . In this paper, using graph-theoretical techniques, we establish an inequality regarding the number of walks and closed walks starting at a given vertex. This inequality yields several upper bounds for the number of closed walks in a graph in terms of the number of vertices, number of edges, maximum degree, degree sequence, the first and the second Zagreb indices of the graph. As applications, in Section 3 we present some new upper bounds on the Estrada index for general graphs, bipartite graphs, trees, and planar graphs, which improve some known results obtained by using the algebraic techniques.
2 Main results
In general, we have the following result.
Each of the equalities holds in (1)-(4) for all v if and only if G is regular.
Proof Let be a walk in . Observe that each of k steps of W has at most Δ choices, then (1) follows. We also notice that the first one, two, and three step(s) of W have exactly , and choices, respectively, and each of the remaining steps has at most Δ choices, so (2), (3), and (4) follow as well. Moreover, it is not difficult to see that each of the equalities holds in (1)-(4) for all v in G if and only if G is a Δ-regular graph. This completes the proof. □
where , are the first and the second Zagreb indices of G, respectively. Using these facts and Lemma 1, for general , we have the following.
Each of the equalities holds in (5)-(8) if and only if G is regular.
Proof This proof is trivial. □
- (i) Let G be a graph of order n with degree sequence . Then for ,(9)
with equality if and only if G is regular or .
- (ii) Let G be a graph of order n with maximum degree Δ. If G admits an orientation with maximum outdegree , then for ,(10)
Moreover, from the proof of (10) (see Theorem 16 in ), one can deduce that the equality holds in (10) if and only if G is a Δ-regular Euler graph.
Now we turn to the number of closed walks. Let denote the set of closed walks of length k starting and ending at v in G, and let . It is obvious that , . In general, for we establish the following simple but useful result.
with equality if and only if k is even, and the component of G containing v is bipartite and v is adjacent to each of the vertices in the other partition part.
Further, the equality holds if and only if the end vertex of each walk in is adjacent to v. In this case, if k is odd then, for any edge vu, contains a walk of the form while v is not adjacent to itself, a contradiction. So if the equality holds in (11), then k must be even.
Now we consider the component of G containing v, under the assumption that the equality holds in (11) and k is even. Let denote the set of vertices at distance l from v in G. We claim that is an empty set, for any . Otherwise there would be a path such that is not adjacent to v, but the walk defined on the path P belongs to , which implies that is adjacent to v, a contradiction. We next show that there are no edges with both end vertices in , . For contradiction, assume that there is an edge, say , with . Then is a triangle and consequently the walk defined on T belongs to , which implies that v is adjacent to v, again a contradiction. Similarly, if there are some edges with both end vertices in , then there must exist a path such that is not adjacent to v, which also yields a contradiction as the above argument. Thus, it follows that the component is bipartite with partition .
The converse is obvious, completing the proof. □
Let denote the number of closed walks of length k in G, i.e., . Clearly, , and . For any , using Lemma 3, we have
with equality if and only if both k and Δ are even and each component of G is the complete bipartite graph .
with equality if and only if k is even, and each component of G is a complete bipartite graph. This result together with bounds (5)-(10) yield bounds (12)-(17) directly; also the equality cases follow by noting that G is Δ-regular (Δ is even in the case of (17)). The proof is completed. □
Recall that an orientation of a graph G is a digraph D obtained from G by choosing an orientation for each edge. The outdegree of a vertex v in D is the number of edges with tail v. It is well known  that a tree (or forest) admits an orientation with maximum outdegree and a planar graph with . In fact, for a forest, fixing a root for each component and orienting each edge in each component toward its root would yield an orientation with ; furthermore, a planar graph has an orientation with since its edges can be partitioned into three forests (see, e.g., ). Thus, by (17) we get an immediate corollary.
Remark that if G is a bipartite graph (including tree and forest), then there are no closed walks of odd length in G, and hence when k is odd. Formally this is stated in the following proposition.
Proposition 6 Let G be a bipartite graph. Then, for any , .
In this section we apply the results in the previous section to estimate the Estrada index of graphs.
We are now ready to give some new upper bounds for .
The discussion for (23)-(27) is analogous by observing that , and, with equality if and only if G is a Δ-regular Euler graph. □
For bipartite graphs, from (21) and the power-series expansion of the hyperbolic cosine , one can easily obtain the following result by a similar reasoning as in the proof of Theorem 7.
with equality if and only if Δ is even and each component of G is the complete bipartite graph .
Similar to Corollary 5, substituting and in (33) and (27), respectively, we have the following corollary.
The authors would like to thank the anonymous referees for their extremely helpful comments and suggestions towards improving the original version of this paper. The first author was supported partially by the Scientic Research Foundation of Guangxi University (Grant No. XBZ130083) and NNSF of China (No. 11361007). The second author was supported by NNSF of China (No. 10831001).
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