A Note on Some Bounds of the α-Estrada Index of Graphs

Let G be a simple graph with n vertices. Let ~ AαðGÞ = αDðGÞ + ð1 − αÞAðGÞ, where 0 ≤ α ≤ 1 and AðGÞ and DðGÞ denote the adjacency matrix and degree matrix of G, respectively. EEαðGÞ =∑i=1ei is called the α-Estrada index of G, where λ1,⋯, λn denote the eigenvalues of ~ AαðGÞ. In this paper, the upper and lower bounds for EEαðGÞ are given. Moreover, some relations between the α-Estrada index and α-energy are established.

In [28], Guo and Zhou proposed the α-Estrada index as where λ 1 , ⋯, λ n are eigenvalues ofÃ α ðGÞ. Obviously, EE 0 ðGÞ is the Estrada index; note that EE 1/2 is somewhat different from the signless Laplacian Estrada index, which is defined to be SLEEðGÞ = ∑ n i=1 e 2λ i , where λ i are the eigenvalues of A 1/2 ðGÞ.
The paper is organized as follows: In Section 2, some bounds for EE α ðGÞ are obtained in terms of the number of vertices, edges, and triangles of G. We also give some new bounds for EE α ðGÞ through different numerical inequalities. Furthermore, some relations between the α-Estrada index and α-energy are established. In Section 3, we compare our new bounds to the existing results for the α-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.

Some Bounds for the α-Estrada Index
In what follows, let trðMÞ denote the trace of matrix M. Let d i denote the degree of vertex i. Lemma 1 (see [1,5]). Let G be a graph with m edges and t triangles. Then In this section, let τðGÞ = ffi, j, kg ⊂ VðGÞ: i, j, k form a triangle of Gg; let β and ξ denote the numbers of subgraphs of G which are isomorphic to path P 3 and cycle C 4 , respectively.

Proposition 2.
Let G be a graph with m edges. Then Proof. Since trðABÞ = trðBAÞ for any A, B ∈ ℝ n×n , we have It is known that trðA 4 ðGÞÞ = 2m + 4β + 8ξ (see [29]). Let A ≔ αDðGÞ and B ≔ ð1 − αÞAðGÞ Taking the trace ofÃ 4 α ðGÞ, we have In the following, we give a lower bound for the α-Estrada index of a graph by using the parameter α, the vertex number, the edge number, and the numbers of subgraphs of G. Theorem 3. Let G be a graph with n vertices, m edges, and t triangles. Then where γ = ∑ fi,jg∈EðGÞ d i d j , ζ = ∑ fi,j,kg∈τðGÞ ðd i + d j + d k Þ.
Proof. By defining EE α ðGÞ, we have According to the Hölder inequality, we have 2m = ∑ i∈VðGÞ d i ≤ n ðt−1Þ/t ð∑ i∈VðGÞ d t i Þ 1/t for any positive integer t.

Corollary 4.
Let G be an r-regular graph with n vertices and t triangles. Then Proof. Since G is an r-regular graph, then m = ð1/2Þnr, Also, we give another lower bound for the α-Estrada index of a graph including the parameter α, the vertex number, the edge number, and the numbers of triangles of G.
Theorem 5. Let G be a graph with n vertices, m edges, and t triangles. Then Proof. Let λ 1 , ⋯, λ n be eigenvalues ofÃ α ðGÞ. By the Taylor expansion theorem, then with equality if and only if x = 0.
By (3), we have By (4) and (11), we have By (5) and (11), we have Summarize the above conclusions, we have In order to prove Theorem 8, we give two lemmas as follows: Lemma 6 (see [27]). Let x 1 , x 2 , ⋯, x n be nonnegative real numbers, and k > 2, then Lemma 7 (see [30]). Let G be a graph with n vertices and m edges. Then Inspired by literatures [14,18], we obtained some bounds on the α-Estrada index by arithmetic-geometric inequality.
Theorem 8. Let G be a graph with n vertices and m edges. Then where Proof. Let λ 1 , ⋯, λ n be eigenvalues ofÃ α ðGÞ. Then By the arithmetic-geometric inequality, we have By the Taylor expansion theorem, we have Advances in Mathematical Physics By substituting the above formula and solving for EE α ðGÞ, we obtain It is elementary to show for n ≥ 2; let the function where 0 ≤ α ≤ 1; then f ðxÞ monotonically decrease in the interval ½0, 4. Let x = 0; f ðxÞ is max; that is to say, ι = 0, EE α ðGÞ is a better lower bound. By Lemmas 6 and 7, we have where ω = In what follows, let λ 1 and λ n be the largest and the smallest ofÃ α ðGÞ, respectively. Lemma 9 (see [1]). Let G be a graph on n vertices with m edges. Then The equality holds if and only if G is a regular graph.
Theorem 10. Let G be a graph on n vertices with m edges. Then Proof.
Consider the function Obviously, the function f ðxÞ is decreasing in x ∈ ð0, 1 and increasing in x ∈ ½1,+∞Þ; then f ðxÞ ≥ f ð1Þ = 0, implying that The equality holds if and only if x = 1. By Lemma 1, we have Define another function Clearly, this is an increasing function on x ∈ ð0,+∞Þ. On the other hand, by Lemma 9, Then, Finally, we get From Theorem 10, we have the following result.
Corollary 11. Let G be a r-regular graph with n vertices. Then

Advances in Mathematical Physics
In the following, we also obtained some other bounds for the α-Estrada index through Sarasija's [31], Ozeki's [32], Polya's [33], and Guo's [34] inequalities, respectively. Lemma 12 (see [31]). Let x 1 , x 2 , ⋯, x n be nonnegative real numbers. Then Theorem 13. Let G be a graph on n vertices with m edges. Then Proof. By Lemma 1 and Lemma 12, let x i = e λ i (i = 1, 2, ⋯, n); we have Then Consider the left and right sides of inequality, respectively, we have Similarly, Lemma 14 (see [32]). If a i and b i are positive real numbers for where M 1 = max 1≤i≤n a i , M 2 = max 1≤i≤n b i , m 1 = min 1≤i≤n a i , and m 2 = min 1≤i≤n b i .

Theorem 15.
Let G be a graph on n vertices with m edges. Then Equality holds if and only if G ≅ K n .
Proof. Let a i = e λ i and b i = 1; then m 1 = e λ n , M 1 = e λ 1 , and m 2 = M 2 = 1, respectively. According to Lemma 14, we have Then Lemma 16 (see [33]). Suppose a i and b i are positive real numbers for 1 ≤ i ≤ n; then where M 1 = max 1≤i≤n a i , M 2 = max 1≤i≤n b i , m 1 = min 1≤i≤n a i , m 2 = min 1≤i≤n b i .

Advances in Mathematical Physics
Theorem 17. Let G be a graph on n vertices with m edges. Then Proof. Let a i = e λ i and b i = 1; then m 1 = e λ n , M 1 = e λ 1 , and m 2 = M 2 = 1, respectively. By Lemma 16, we have Then Lemma 18 (see [34]). For a 1 , a 2 , ⋯, a n ≥ 0 and p 1 , where T = min fp 1 , p 2 , ⋯, p n g. Equality holds if and only if a 1 = a 2 = ⋯ = a n .

Theorem 19.
Let G be a graph with n vertices with m edges. Then where Δ = e ð2αmð2n−1Þ−nλ 1 Þ/ð2nðn−1ÞÞ . Equality holds if and only if G ≅ K n .
In the H€ uckel 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 α-energy of G is defined as ς α ðGÞ = ∑ n i=1 jλ i − ð2am/nÞj, where λ 1 , ⋯, λ n are the eigenvalues of A α ðGÞ. New bounds for the α-Estrada index in terms of the α-energy of the graph G are established.
In [37], the Estrada index-like quantity is defined by where x 1 , x 2 , ⋯, x n are arbitrary real numbers and x is their arithmetic mean. Let x 1 , x 2 , ⋯, x n and x be λ 1 , ⋯, λ n and 2 am/n, respectively. Evidently, EE α ðGÞ = e 2am/n EE α ðGÞ, and therefore, results obtained for EE α ðGÞ can be immediately restated for EE α ðGÞ and vice versa.

Theorem 21.
Let G be a graph on n vertices with m edges. Then Proof. Let x ≥ 0, considering the following function: in which equality holds if and only ifx = 0. The function f ðxÞ is increasing in ½0, +∞Þ. The f ðxÞ ≥ f ð0Þ, implying that By (70), we have Theorem 22. Let G be a graph on n vertices with m edges. Then in which equality holds if and only ifδ 1 = ⋯ = δ k and δ k+1 = ⋯ = δ n .
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 e δ 1 = ⋯ = e δ k and e δ k = ⋯ = e δ n ; that is to say, the equality holds in (75) if and only if δ 1 = ⋯ = δ k and δ k+1 = ⋯ = δ n . This means all negative eigenvalues and all nonnegative eigenvalues which completes the proof.

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 EE 0 ðGÞ (see Table 1).
The C 20 , C 40 , and C 60 are fullerenes (letter C is followed by the number of carbon atoms). ERð1Þ is the Erdös-Rényi random graph with n = 100 and p = 0:05. ERð2Þ is the Erdös-Rényi random graph with n = 100 and p = 0:5. BA is the Barabási-Albert random graph with n = 100, m = 5, and n 0 = 50. WSð1Þ is the Watts-Strogatz random graph with n = 100, K = 6, and p = 0:1. WSð2Þ is the Watts-Strogatz random graph with n = 100 and K = 6, and p = 0:5. GN is the GN (Girvan-Newman) Benchmark graph with n = 128, k = 16, max k = 16, mu = 0:1, min c = 32, and max c = 32. LFR is the LFR (Lancichinetti-Fortunato-Radicchi) Benchmark graph with n = 1000, k = 10, max k = 40, mu = 0:2, min c = 30, and max c = 60 (for related parameters, see 8 Advances in Mathematical Physics [39,40]). We use f x instead of 10 x 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. 2:13 − are better than the lower bound of Th. 1; the upper bound of Th. 2:13 + is better than the upper bound of Th. 2.1. We also get some other results in Table 1

Conclusion
In this paper, we give some bounds on the α-Estrada index of G, some relations between the α-Estrada index and α-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 α-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][8][9][10][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),