Abstract
Let be a simple graph with vertices. Let , where and and denote the adjacency matrix and degree matrix of , respectively. is called the -Estrada index of , where denote the eigenvalues of . In this paper, the upper and lower bounds for are given. Moreover, some relations between the -Estrada index and -energy are established.
1. Introduction
Let be a simple undirected graph with vertices, where denotes the vertex set of and denotes the edge set of . Let and denote the adjacency matrix and degree matrix of , respectively. The Laplacian (signless Laplacian) matrix of is . In [1], Nikiforov defined and studied some problems of (for example, spectral extremal problems). Some results on have been obtained, including bounds of the eigenvalue of and the positive semidefiniteness of [1, 2], etc. For more spectral properties of , see [3–6].
The Estrada index [7] of is defined as where are eigenvalues of . The Estrada index can be used to measure the folding degree of long-chain proteins [8, 9] and subgraph centrality in complex networks [10–13].
The Laplacian Estrada index and signless Laplacian Estrada index of are defined as and , respectively, where and are eigenvalues of and , respectively [14, 15]. Some mathematical and chemical properties of , , and are investigated extensively in mathematical chemistry [14, 16–25]. For other generalized Estrada index, see [26, 27].
In [28], Guo and Zhou proposed the -Estrada index as where are eigenvalues of . Obviously, is the Estrada index; note that is somewhat different from the signless Laplacian Estrada index, which is defined to be , where are the eigenvalues of .
The paper is organized as follows: In Section 2, some bounds for are obtained in terms of the number of vertices, edges, and triangles of . We also give some new bounds for through different numerical inequalities. Furthermore, some relations between the -Estrada index and -energy are established. In Section 3, we compare our new bounds to the existing results for the -Estrada index by certain graphs, benchmark graphs, and random graphs. In Section 4, we summarize the results of the paper, and the future work is envisaged.
2. Some Bounds for the -Estrada Index
In what follows, let denote the trace of matrix . Let denote the degree of vertex .
Lemma 1 (see [1, 5]). Let be a graph with edges and triangles. Then
In this section, let ; let and denote the numbers of subgraphs of which are isomorphic to path and cycle , respectively.
Proposition 2. Let be a graph with edges. Then
Proof. Since for any , we have It is known that (see [29]). Let and Taking the trace of , we have
In the following, we give a lower bound for the -Estrada index of a graph by using the parameter , the vertex number, the edge number, and the numbers of subgraphs of .
Theorem 3. Let be a graph with vertices, edges, and triangles. Then where .
Proof. By defining , we have According to the Hölder inequality, we have for any positive integer . Hence By (3)–(11) and Proposition 2, we have where , .
Corollary 4. Let be an -regular graph with vertices and triangles. Then
Proof. Since is an -regular graph, then , , . By Theorem 3, we have
Also, we give another lower bound for the -Estrada index of a graph including the parameter , the vertex number, the edge number, and the numbers of triangles of .
Theorem 5. Let be a graph with vertices, edges, and triangles. Then
Proof. Let be eigenvalues of . By the Taylor expansion theorem, then with equality if and only if . By (3), we have So By (4) and (11), we have So By (5) and (11), we have So Summarize the above conclusions, we have
In order to prove Theorem 8, we give two lemmas as follows:
Lemma 6 (see [27]). Let be nonnegative real numbers, and , then
Lemma 7 (see [30]). Let be a graph with vertices and edges. Then Inspired by literatures [14, 18], we obtained some bounds on the -Estrada index by arithmetic-geometric inequality.
Theorem 8. Let be a graph with vertices and edges. Then where
Proof. Let be eigenvalues of . Then
By the arithmetic-geometric inequality, we have
By the Taylor expansion theorem, we have
Let , we have
By substituting the above formula and solving for , we obtain
It is elementary to show for ; let the function
where ; then monotonically decrease in the interval . Let ; is max; that is to say, , is a better lower bound.
By Lemmas 6 and 7, we have
where
In what follows, let and be the largest and the smallest of , respectively.
Lemma 9 (see [1]). Let be a graph on vertices with edges. Then The equality holds if and only if is a regular graph.
Theorem 10. Let be a graph on vertices with edges. Then
Proof. Consider the function Obviously, the function is decreasing in and increasing in ; then , implying that The equality holds if and only if . By Lemma 1, we have Define another function Clearly, this is an increasing function on . On the other hand, by Lemma 9, Then, Finally, we get
From Theorem 10, we have the following result.
Corollary 11. Let be a -regular graph with vertices. Then
In the following, we also obtained some other bounds for the -Estrada index through Sarasija’s [31], Ozeki’s [32], Polya’s [33], and Guo’s [34] inequalities, respectively.
Lemma 12 (see [31]). Let be nonnegative real numbers. Then
Theorem 13. Let be a graph on vertices with edges. Then
Proof. By Lemma 1 and Lemma 12, let (); we have Then Consider the left and right sides of inequality, respectively, we have Similarly,
Lemma 14 (see [32]). If and are positive real numbers for , then where , , , and .
Theorem 15. Let be a graph on vertices with edges. Then Equality holds if and only if .
Proof. Let and ; then , , and , respectively. According to Lemma 14, we have Then
Lemma 16 (see [33]). Suppose and are positive real numbers for ; then where .
Theorem 17. Let be a graph on vertices with edges. Then
Proof. Let and ; then , , and , respectively. By Lemma 16, we have Then
Lemma 18 (see [34]). For and such that . Then where . Equality holds if and only if .
Theorem 19. Let be a graph with vertices with edges. Then where . Equality holds if and only if .
Proof. Let , for , and for . Obviously, , according to Lemma 18; we have
where .
We consider is
Since
then
equality holds that is if and only if .
In the Hckel molecular orbital theory, graph energy is defined as the sum of the absolute values of the eigenvalues of the adjacency matrix of the molecular graph [35, 36]. In [28], the -energy of is defined as , where are the eigenvalues of . New bounds for the -Estrada index in terms of the -energy of the graph are established.
In [37], the Estrada index-like quantity is defined by where are arbitrary real numbers and is their arithmetic mean. Let and be and , respectively. Evidently, , and therefore, results obtained for can be immediately restated for and vice versa.
Theorem 20. Let be a graph on vertices with edges. Then
Proof. Note that . By Lemma 9, we have In order to prove Theorems 21 and 22, let , and let . By Lemma 1, we obtained . The -energy of the graph is ; then .
In the following, some relations between the -Estrada index and -energy are established.
Theorem 21. Let be a graph on vertices with edges. Then
Proof. Let , considering the following function: in which equality holds if and only if. The function is increasing in . The , implying that By (70), we have
Theorem 22. Let be a graph on vertices with edges. Then in which equality holds if and only if and .
Proof. By the Mean Quadratic inequality, we have Similarly, Then
The equality holds in (75) if and only if equalities hold in both (73) and (74). By the equality case in the Mean Quadratic inequality, equality occurs in (73) and (74) if and only if and ; that is to say, the equality holds in (75) if and only if and . This means all negative eigenvalues and all nonnegative eigenvalues which completes the proof.
3. Numerical Examples
In this section, we list some computational experiments to compare our new bounds to previous results for certain connected graphs, benchmark graphs, and random graphs, where the results of the benchmark graphs and random graphs are the average of 20 independent experiments. We listed the lower bound of Theorem 1 (Th. 1) [23], the lower bound of Theorem 10 (Th. 10), the lower bound of Theorem 13 (Th. 13–), the upper bound of Theorem 2.1 (Th. 2.1) [38], the upper bound of Theorem 13 (Th. 13+), and the numerical value of (see Table 1).
The , , and are fullerenes (letter is followed by the number of carbon atoms). is the Erdös-Rényi random graph with and . is the Erdös-Rényi random graph with and . is the Barabási-Albert random graph with , , and . is the Watts-Strogatz random graph with , , and . is the Watts-Strogatz random graph with and , and . is the GN (Girvan-Newman) Benchmark graph with , , , , , and . is the LFR (Lancichinetti-Fortunato-Radicchi) Benchmark graph with , , , , , and (for related parameters, see [39, 40]). We use instead of in Table 1. The results are kept to four decimal places.
According to the information in Table 1, we know that the lower bounds in Th. 2.10 and Th. are better than the lower bound of Th. 1; the upper bound of Th. is better than the upper bound of Th. 2.1. We also get some other results in Table 1 as follows: The lower bound of Th. 2.10 is better than the lower bounds of Th. in the cycle graph, bipartite graph, Petersen graph, , , and ; Th. is good in other cases. For sparse graphs, Th. is good in most cases. For dense graph, Th. is good in most cases.
4. Conclusion
In this paper, we give some bounds on the -Estrada index of , some relations between the -Estrada index and -energy are established. At the same time, we also analyze the advantages and disadvantages of different bounds for certain connected graphs, benchmark graphs, and random graphs by numerical experiments. Our future work will focus on exploring the practical applications of the -Estrada index in physical, chemical, and network sciences.
Data Availability
The Estrada index is a spectral measure to character efficiently the strongness of complex networks. These prior studies (and datasets) are cited at relevant places within the text as references [7–11, 29]. Since the paper is a theoretical study, so no data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (No. 11801115 and No. 11601102), the Natural Science Foundation of Heilongjiang Province (No. QC2018002), and the Fundamental Research Funds for the Central Universities.