The Scientific World Journal

Volume 2014 (2014), Article ID 637865, 7 pages

http://dx.doi.org/10.1155/2014/637865

Research Article

On the Maximum Estrada Index of 3-Uniform Linear Hypertrees

Faxu Li,1,2 Liang Wei,³ Jinde Cao,4,5 Feng Hu,1,2 and Haixing Zhao²

¹School of Computer Science, Shaanxi Normal University, Xi’an 710062, China

²College of Computer, Qinghai Normal University, Xining 810008, China

³Department of Mathematics, Qinghai Normal University, Xining 810008, China

⁴Department of Mathematics, Southeast University, Nanjing 210096, China

⁵Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received 3 July 2014; Accepted 12 August 2014; Published 28 August 2014

Academic Editor: Jianlong Qiu

Copyright © 2014 Faxu Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

For a simple hypergraph on vertices, its Estrada index is defined as , where are the eigenvalues of its adjacency matrix. In this paper, we determine the unique 3-uniform linear hypertree with the maximum Estrada index.

1. Introduction

Let be a simple graph, and let and be the number of vertices and the number of edges of , respectively. The characteristic polynomial of a graph is written as , where is the adjacency matrix of . The eigenvalues of are the eigenvalues of its adjacency matrix , which are denoted by . A graph-spectrum-based invariant, nowadays named Estrada index, proposed by Estrada in 2000, is defined as [1]

Since then, the Estrada index has already found remarkable applications in biology, chemistry, and complex networks [2–5]. Some mathematical properties of the Estrada index, especially bounds for it have been established in [6–15]. For more results on the Estrada index, the readers are referred to recent papers [16–19].

Let be a simple and finite hypergraph with vertex set and hyperedge set . The hypergraph is called linear if two hyperedges intersect in one vertex at most and also -uniform if for each in , . An -uniform hypertree is a connected linear -hypergraph without cycles. An -uniform linear hypertree is called 3-uniform linear hypertree if is equal to 3. Denoted by an -uniform linear star with hyperedges. More details on hypergraphs can be found in [20].

Let denote a square symmetric matrix in which the diagonal elements are zero, and other elements represent the number of hyperedges containing both vertices and (for undirected hypergraphs, ). Let be the eigenvalues of of . The subhypergraph centrality of a hypergraph , firstly put forward by Estrada and Rodríguez-Velázquez in 2006, is defined as [21]

They revealed that the subhypergraph centrality provides a measure of the centrality of complex hypernetworks (social, reaction, metabolic, protein, food web, etc). For convenience, we call the subhypergraph centrality of a hypergraph its Estrada index and define the Estrada index as

Thus far, results on the Estrada index of hypergraph seem to be few although the Estrada index of graph has numerous applications. So our main goal is to investigate the Estrada index of 3-uniform linear hypertrees. In this paper, we determine the unique 3-uniform linear hypertree with the maximum Estrada index among the set of 3-uniform linear hypertrees.

2. Preliminaries

For a hypergraph of order , its completely connected graph, denoted by , is a graph which has the same order and in which two vertices are adjacent if they share one hyperedge. Obviously, is a multigraph. For an -uniform linear hypergraph , is a simple graph. According to the definition of adjacency matrix of hypergraph, it is easy to see that both a 3-uniform linear hypertree and its completely connected graph have the same adjacency matrix; see Figure 1. Then, they have the identical Estrada index. Thus, we investigate the Estrada index of its completely connected graphs instead of the 3-uniform linear hypertrees in this paper.

Figure 1: A 3-uniform linear hypertree and its completely connected graph .

We use to denote the th spectral moment of the graph . It is well-known [22] that is equal to the number of closed walks of length in . Obviously, for any graph , , , , , and , where , , and are the number of triangles, the number of quadrangles, and the degree of vertex in graph , respectively. Then

For , denote by the set of -walks of length in . Obviously, . For convenience, let and . Let be a -walk in graph ; we denote by a -walk obtained from by reversing .

For any two graphs and , if for all integers , then . Moreover, if the strict inequality holds for at least one value , then .

Denote by the set of connected graphs on vertices and triangles such that any two triangles have a common vertex at most. Apparently, for a 3-uniform linear hypertree on vertices and hyperedges, . Now we study the Estrada index of a graph in .

3. Maximum Estrada Index of 3-Uniform Linear Hypertrees

In this section, we determine the maximum value of Estrada index among the set of 3-uniform linear hypertrees.

Lemma 1. Let be star which is the completely connected graph of with hyperedges. It is easily found that the star has vertices labled and triangles. Let be a positive integer; then there is an injection from to , and is not surjective for , , and , where and are the sets of closed walks of length of and in , respectively; see Figure 2.

Figure 2: The star on triangles.

Proof. Firstly, we construct a mapping from to . For , let be the closed walk obtained from by replacing by and by . Obviously, and is a bijection.

Secondly, we construct a mapping from to . For , we consider the following cases.

Case  1. Suppose does not pass the edge for ; then .

Case  2. Suppose passes the edge for . For , we may uniquely decompose into three sections , where is the longest -section of without , is the internal longest -section of for , and the last is the remaining -section of not containing . We consider the following three subcases.

Case  2.1. If both and contain the vertex , we may uniquely decompose into two sections and decompose into two sections , where is the shortest -section of , is the remaining -section of , is the longest -section of , and is the remaining -section of .

Let , where = , , , is a -walk obtained from replacing by and by , and is a -walk obtained from replacing by and by .

Case  2.2. If contains the vertex and does not contain , let , where , is a -walk obtained from replacing its first vertex by , and is a -walk obtained from replacing its last two vertices by .

Case  2.3. If does not contain the vertex , let , where , is a -walk obtained from replacing its first two vertices by , and is a -walk obtained from replacing its last vertex by .

For example, in star on vertices and triangles, is a closed walk of length 6 of not passing the edge . By Case  1, we have

is a closed walk of length 9 of passing the edge . By Case  2.2, we get

is a closed walk of length 14 of passing the edge . By Case  2.3, we obtain

Obviously, , is an injective and not a surjective for , and .

Lemma 2. Let be a nonisolated vertex of a connected graph . If and are the graphs obtained from by identifying an external vertex and the center vertex of the union of to , respectively, where , is either empty graph or nonempty graph. Then for and ; see Figure 3.

Figure 3: Transformation .

Proof. Let , resp.) be the set of closed walks of length of , , resp.) for . Then is a partition, where is the set of closed walks of length of ; each of them contains both at least one edge in and at least one edge in . So . Thus we need to show the inequality .

We construct a mapping from to and consider the following four cases.

Case  1. Suppose is a closed walk starting from in . For , let ; that is, is the closed walk in obtained from by replacing its every section in with its image under the map .

Case  2. Suppose is a closed walk starting at in . For , we may uniquely decompose into three sections , where is the longest -section of without vertices , is the internal longest -section of (for which the internal vertices are some possible vertices in ), and is the remaining -section of . Let , where , , and ; that is, is a -walk from by replacing its every section in with its image under the map .

Case  3. Suppose is a closed walk starting from or in . For , we may uniquely decompose into three sections , where is the longest (or )-section of without vertices , is the internal longest -section of (for which the internal vertices are some possible vertices in ), and is the remaining (or )-section of without vertices . We have three subcases.

Case  3.1. If both and do not pass edge , let , where , is a (or )-walk obtained from replacing by and by , and is a (or )-walk obtained from replacing by and by .

Case  3.2. If both and pass edge , we may anew decompose into five sections , where is the longest (or )-section of (which do not contain vertices ), is the second -section of (for which the internal vertices, if exist, are only possible , ), the third is the internal longest -section of (for which the internal vertices are some possible vertices in ), the fourth is the longest -section of (for which the internal vertices, if exist, are only possible , ), and the last is the remaining (or )-section of . We have three subsubcases.

Case  3.2.1. If both and contain the vertex , we may uniquely decompose into two sections and into two sections , where is the longest -section of , is the remaining shortest of , is the shortest -section of , and is the remaining longest -section of .

Let , where = , = , , = = , is a -walk obtained from replacing by and by , and is a -walk obtained from replacing by and by .

Case  3.2.2. If does not contain the vertex , let , where , , , is a -walk obtained from replacing its last two vertices by , and is a -walk obtained from replacing its first vertex by .

Case  3.2.3. If contains the vertex and does not contain vertex , let , where , = , , is a -walk obtained from replacing its last vertex by , and is a -walk obtained from replacing its first two vertices by .

Case  3.3. If passes edge and does not pass edge , we may anew decompose into four sections , where is the longest (or )-section of (which do not contain vertices ), is the second -section of (for which the internal vertices, if exist, are only possible ), the third is the internal longest -section of (for which the internal vertices are some possible vertices in ), and the last is the longest (or )-section of (for which the internal vertices, if exist, are only possible , ). We consider the following two subsubcases.

Case  3.3.1. If contains vertex , we may uniquely decompose into two sections , where is the longest -section of and is the remaining shortest -section of .

Let , where , , = , is a -walk obtained from replacing by and by , and is a (or )-walk obtained from replacing by and by .

Case  3.3.2. If does not contain vertex , let , where , , is a -walk obtained from replacing its last two vertices by , and is a (or )-walk obtained from replacing its first vertex by .

Case  3.4. If does not pass edge and passes edge , we may anew decompose into four sections , where is the longest (or )-section of (which do not contain vertices and must contain vertex ), the second is the internal longest -section of (for which the internal vertices are some possible vertices in ), is the third -section of (for which the internal vertices, if exist, are only possible ), and the last is the longest (or )-section of . We have two subsubcases.

Case  3.4.1. If contains vertex , we may uniquely decompose it into two sections , where is the shortest -section of and is the remaining longest -section of .

Let , where , , , is a (or )-walk obtained from replacing by and by , and is a -walk obtained from replacing by and by .

Case  3.4.2. If does not contain vertex , let , where , , is a (or )-walk obtained from replacing its last vertex by , and is a -walk obtained from replacing its first two vertices by .

Case  4. Suppose is a closed walk starting from for in . For , we may uniquely decompose into five sections , where is the longest -section of (which do not contain vertices ), is the second -section of (for which the internal vertices, if exist, are only possible ), the third is the internal longest -section of (for which the internal vertices are some possible vertices in ), the fourth is the longest -section of (for which the internal vertices, if exist, are only possible ), and the last is the remaining -section of . We have four subcases.

Case  4.1. If both and contain the vertex , we may uniquely decompose into two sections and decompose into two sections , where is the longest -section of , is the remaining shortest of , is the shortest -section of , and is the remaining longest -section of .

Let , where = , = , , , is a -walk obtained from replacing by and by , and is a -walk obtained from replacing by and by .

Case  4.2. If contains the vertex and does not contain vertex , let , where = , , , is a -walk obtained from replacing its last vertex by , and is a -walk obtained from replacing its first two vertices by .

Case  4.3. If does not contain the vertex , let , where , , , is a -walk obtained from by replacing its last two vertices by , and is a -walk obtained from by replacing its first vertex by .

For example,

where , , are vertices in and are vertices in .

By Lemma 1, is injective and not surjective. It is easily shown that is also injective and not surjective. Thus , .

Theorem 3. Let be an arbitrary graph on vertices in set , where . Then with the equality holding if and only if .

Proof. Determine a vertex of the maximum degree as a root in , and let be an integer. Let be the completely connected graph of 3-uniform linear hypertree attached at , and let be the number of triangles of for , respectively. We can repeatedly apply this transformation from Lemma 2 at some vertices whose degrees are not equal to two or in till becomes a star. From Lemma 2, it satisfies that each application of this transformation strictly increases the number of closed walks and also increases Estrada index.

When all turn into stars, we can again use Lemma 2 at the vertex as long as there exists at least one vertex whose degree is not equal to two or , further increasing the number of closed walks. In the end of this procedure, we get the star . The whole procedure of transformation is shown in Figure 4.

Figure 4: The procedure of transformation.

Lemma 4 (see [20]). Let be a vertex of a graph , for , and the set of cycles containing . Consider

where if is a single edge and if is a cycle.

Now, we calculate . Applying Lemma 4, we have

By some simple calculating, we achieve the following eigenvalues:

Then, we obtain

Theorem 3 shows that the star has the maximum Estrada index in set . Thus, according to previous definition, it is easy to show that the 3-uniform star has the maximum Estrada index among the set of 3-uniform linear hypertrees; that is,

where

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT)(no. ITR1068), Special 973 Program for Key Basic Research of the Ministry of Science and Technology, China (no. 2010CB334708), the National Natural Science Foundation of China (NSFC)(no. 60863006), Scientific Research Foundation of the Department of Science and Technology, Qinghai Province, China (no. 2012-Z-943).

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