On the Estrada Indices of Unicyclic Graphs with Fixed Diameters

: The Estrada index of a graph G is deﬁned as EE ( G ) = ∑ ni = 1 e λ i , where λ 1 , λ 2 , . . . , λ n are the eigenvalues of the adjacency matrix of G . A unicyclic graph is a connected graph with a unique cycle. Let U ( n , d ) be the set of all unicyclic graphs with n vertices and diameter d . In this paper, we give some transformations which can be used to compare the Estrada indices of two graphs. Using these transformations, we determine the graphs with the maximum Estrada indices among U ( n , d ) . We characterize two candidate graphs with the maximum Estrada index if d is odd and three candidate graphs with the maximum Estrada index if d is even.


Introduction
In this paper, we only consider simple undirected graphs. Let G = (V(G), E(G)) be a graph with n vertices and m edges. Let N G (v) be the set of vertices adjacent to v in G. The degree of v in G, denoted by d G (v), is equal to |N G (v)|. A vertex of degree one is called a pendant vertex. The edge incident with a pendant vertex is known as a pendant edge. Let S = ∅ ⊆ V(G). Then denote by G[S] the subgraph induced by S. If D ⊆ E(G) (or D ⊆ V(G)), then we write G − D for the graph obtained from G by deleting all of its edges (or vertices, resp.) in D. If D ⊆ E(G), then we denote by G + D the graph obtained from G by adding all of edges in D to the graph.
Let A(G) be the adjacency matrix of G. Denote the eigenvalues of A(G) by λ 1 , λ 2 , . . . , λ n and assume λ 1 ≥ λ 2 ≥ . . . ≥ λ n . Then λ 1 , usually denoted by ρ(G), is called the spectral radius of G. The Estrada index of G is defined as This graph invariant was first proposed as a measure of the degree of folding of a protein [1] and now has been found multiple applications in various fields, such as measurements of the subgraph centrality and the centrality of complex networks [2,3] and the extended molecular branching [4]. Recently, the correlation between the Estrada index and π-electronic energies for benzenoid hydrocarbons was investigated in [5], the results of which warrant its further usage in quantitative structure-activity relationships. Given these prominent applications of the Estrada index, the research on it is of theoretical and practical significance. In the last few decades, some mathematical properties of the Estrada index, including various bounds for it, have been established [6][7][8][9][10][11][12].
In 1986, Brualdi and Solheid [13] proposed the following problem concerning the spectral radii of graphs: Given a set G of graphs, find an upper bound for the spectral radius of graphs in G and characterize the graphs for which the maximal spectral radius is attained. The corresponding problem of a given graph invariant has been widely studied (see [14][15][16], for example). Motivated by this, many results have been obtained on characterizing graphs that maximize (or minimize) the Estrada index among a given set of graphs. For example, some interesting results were obtained for the general trees [17], trees with a given matching number [18], trees with a fixed diameter [19], trees with perfect matching and a fixed maximum degree [20], and trees with a fixed number of pendant vertices [21]. Du and Zhou [22] showed a graph with the maximal Estrada index and two candidate graphs with the minimum Estrada index among all unicyclic graphs. Moreover, they determined the unique graphs with maximum Estrada indices among graphs with given parameters [23]. Wang et al. [24] and Zhu et al. [25] characterized the bicyclic graph and the tricyclic graph with maximum Estrada indices, respectively. E. Andrade et al. [26] presented the graph having the largest Estrada index of its line graph among all graphs on n vertices with connectivity less than or equal to a fixed number. For more results on the Estrada index and its variations, the readers may refer to [27][28][29][30].
A unicyclic graph is a connected graph with a unique cycle. Let P n and C n be the path and the cycle on n vertices, respectively. Denote by U (n, d) the set of all unicyclic graphs with n vertices and diameter d. In this paper, we characterize the graphs with the maximum Estrada index in U (n, d).
This paper is organized as follows. In Section 2, we list some transformations which can be used to compare the Estrada indices of two graphs. In Section 3, we determine the graphs with the maximum Estrada index among unicyclic graphs in U (n, d). We show two candidate graphs with the maximal Estrada index if d is odd and three candidate graphs with the maximal Estrada index if d is even. We also propose a hypothesis on the structure of the extremal graph with the maximum Estrada index in U (n, d).

Preliminaries
In order to obtain the main results of this paper, we give some definitions and lemmas here.
A walk of length k in a graph G is any sequence of vertices and edges in G, W = v 1 e 1 v 2 e 2 · · · v k e k v k+1 , such that e i = v i v i+1 for every 1 ≤ i ≤ k. For a subsequence v i e i v i+1 · · · v j−1 e j−1 v j of W, we refer to it as a (v i , v j )-section of W. Usually, we write Let M k (G) be the kth spectral moment of the graph G defined as M k (G) = ∑ n i=1 λ k i . It is well-known that M k (G) equals the number of closed walks of length k in G; see [31]. Then by the Taylor expansion of the exponential function e x , we have Let G and H be two graphs with x, y ∈ V(G), u, v ∈ V(H) and e ∈ E(G). Suppose k is an arbitrary positive integer. Let W k (G; x, [e]) be the set of all (x, x)-walks of length k going through the edge e in G and let |W k (G; x, [e])| = M k (G; x, [e]). Let W k (G; x, y) be the set of all (x, y)-walks of length k in G and let |W k (G; x, y)| = M k (G; x, y). If M k (G; x, y) ≤ M k (H; u, v) for all positive integers k, then we write (G; x, y) (H; u, v). If (G; x, y) (H; u, v), and M k 0 (G; x, y) < M k 0 (H; u, v) for some positive integer k 0 , then we write (G; x, y) ≺ (H; u, v). For convenience, let W k (G; x) = W k (G; x, x), M k (G; x) = M k (G; x, x) and (G; u) = (G; u, u).
The following four results are often used to compare the Estrada indices of two graphs.  ([32]). Let H 1 and H 2 be two non-trivial graphs with u, v ∈ V(H 1 ), w ∈ V(H 2 ). Let G u be the graph obtained from H 1 and H 2 by identifying u with w, and G v be the graph obtained from H 1 and H 2 by identifying v with w. If (H 1 ; v) ≺ (H 1 ; u), then EE(G v ) < EE(G u ).

Lemma 3 ([32]
). Let G 1 and G 2 be two connected graphs with u ∈ V(G 1 ) and v ∈ V(G 2 ). Let G be the graph obtained by joining u and v with and edge, and let G be the graph obtained by identifying u with v, and attaching a pendant vertex to the common vertex. If d G (u), d G (v) ≥ 2, then EE(G) < EE(G ). Theorem 1 ([27]). Let G be a connected graph and G u,v (p, q) be the graph obtained from G by attaching p and q pendant edges to u and v, respectively, where u, v ∈ V(G) and p, q ≥ 1.

Lemmas
In this section, we give some lemmas that can be used to prove

Lemma 5. Let G be a graph and H
Proof of Lemma 5. For each z ∈ {u, v} and k ≥ 0, by the definition of M k (H; z), ). Then either veu or uev must be contained in W. If W does not contain the section uev, or veu appears earlier than uev in W, then W can be decomposed ). If W does not contain the section veu, or uev appears earlier than veu in W, then W can be decomposed uniquely to Here, W r either contains no e, or contains no uev, or veu appears earlier than uev. Without loss of generality, we suppose veu appears earlier than uev in W r . Then W r can be decomposed to W r = W t+1 eW t+2 such that W t+1 ∈ W k t+1 (G; v) for some k t+1 ≥ 0 and W t+2 is a (u, v)-walk in H. In this case, we . Now it is easy to show that the map h k : Lemma 6. Let G be a graph and P = v 0 v 1 · · · v m be a path in G such that d G (v 0 ) = 1. Let q and l be two nonnegative integers such that 0 ≤ q < l and q + l ≤ m. Suppose v = v q and u = v l are two vertices in P such that d G ( (2) If q + l < m, or q + l = m and the condition C does not hold, then , denote by W the walk obtained from W by replacing each vertex v x with v x and the corresponding edges, where x = q + l − x. We distinguish the following two cases: has the same distance from v and u in P.
If W contains v a more than once, then it can be decomposed uniquely to k does not cover the walk v l v l+1 · · · v m−1 v m v m−1 · · · v l+1 v l . Now suppose q + l = m and the condition C does not hold. Without loss of generality, suppose there exists some a does not cover the walk v l v l+1 · · · v j sv j · · · v l+1 v l . Therefore, if q + l < m, or q + l = m and the condition C does not hold, then M k 0 (G; v) < M k 0 (G; u) for some k 0 ≥ 0. This implies Lemma 6 (2) holds.
Let w ∈ V(G)\{v 0 , v 1 , . . . , v a−1 } and W ∈ W k (G; w, v). Then W must contain v a . Thus, W can be decomposed uniquely to W = W 1 W 2 such that W 1 ∈ W k 0 (G; w, v a ) which is as long as possible and W 2 is a (v a , v)-walk in G. Then the map g k : W k (G; w, v) → W k (G; w, u) defined as g k (W) = W 1 W 2 is an injection. Therefore, (G; w, v) (G; w, u).
Case 2. q + l is odd. Let k ≥ 0, W ∈ W k (G; v), and e a = v a v a+1 . If W contains e a , then it must contain e a at least twice and can be decomposed uniquely to W = W 1 W 2 e a W 3 e a W 4 , such that W 1 ∈ W k 1 (G; v, v a ), which contains v a only once; W 2 ∈ W k 2 (G; v a ), which does not contain e a ; W 3 ∈ W k 3 (G; v a+1 ), which is as long as possible; and W 4 ∈ W k 4 (G; v a , v), which does not contain e a . In this case, let f (2) k (W) = W 1 W 3 e a W 2 e a W 4 . If W does not contain e a , let f The proof of (Case 2) when q + l is odd is the same as that of Case 1.

Graphs with the Maximum Estrada Index in U (n, d)
In this section, we determine the graphs with the maximum Estrada index among U (n, d).
The following theorem characterizes the graphs with greatest, second-greatest, smallest, and second-smallest Estrada indices among the unicyclic graphs in U (n).
For G ∈ U (n, d), we have n ≥ 3 and 1 ≤ d ≤ n − 2. If d = 1, then G = C 3 . By Theorem 2, the graphs with the maximum Estrada indices among the graphs in U (n, 2) and U (n, 3) are X n and C 3 (0, 1, n − 4), respectively. Therefore, we assume d ≥ 4 and n ≥ 6 in the following. Now we give some lemmas.

Proof of Lemma 7. Let
) which is as long as possible, W 2 = e d 2 −1 or ev d+1 e, W 4 = e d 2 −1 or ev d+1 e. Obviously, neither W 1 nor W 5 contains v d 2 . In this case, we define f k (W) = W 3 W 4 W 5 W 2 W 1 . Then it is easy to show that the map f k : defined as above is an injection. Since f k does not cover the walk v d ).
Lemma 8. Let C 4 = xvyux be a 4-cycle. Denote by H the graph obtained from C 4 by attaching two paths P p = v 1 v 2 · · · v p and P q = u 1 u 2 · · · u q at vertices v and u, respectively; see Figure 2. If 0 ≤ p < q, then (H; v) ≺ (H; u).  that the map f k : W k (H; v) −→ W k (H; u) defined as above is an injection. Since p < q, f k does not cover the walk uu 1 · · · u q−1 u q u q−1 · · · u 1 u. Therefore, we have (H; v) ≺ (H; u).
Proof of Lemma 9. Let k ≥ 0 be an arbitrary integer. By the definition of the walk, we have and . We consider the edge e preceding the last vertex v in W. If e = (x, v), then W can be written as Define W 2 the walk obtained from W 2 by replacing v i with u i for each 1 ≤ i ≤ p, v with u, and the corresponding edges. By the inductive hypothesis, there is an injection f k 1 : W k 1 (H; x, v) → W k 1 (H; x, u). In this case, let f k (W) = f k 1 (W 1 )W 2 . Then it is easy to show that the map f k : W k (H; x, v) −→ W k (H; x, u) defined as above is an injection. Therefore, M k (H; x, v) ≤ M k (H; x, u).
Let P = v 0 v 1 · · · v d be a path of length d with d ≥ 2. Let ∆ d n be the graph obtained from P and a new vertex v d+1 by adding the edges v d 2 v d+1 and v d 2 +1 v d+1 , and attaching n − d − 2 pendant edges at the vertex v d 2 (see Figure 3). Lemma 10. Let P = v 0 v 1 · · · v d be a path of length d ≥ 4. Let G v k ,v be the graph with diameter d obtained from P and a new vertex v d+1 by adding the edges v k v d+1 and v k+1 v d+1 , and attaching where ∆ d n is depicted in Figure 3.
Proof of Lemma 10. Denote by B the set of all graphs G v k ,v . Let G * be the graph in B with the maximum Estrada index. Then there exists some 0 ≤ k ≤ d − 1 such that G * is obtained from P and v d+1 by adding the edges v k v d+1 and v k+1 v d+1 , and attaching n − d − 2 pendant edges at a vertex v for some vertex v ∈ V(P) ∪ {v d+1 }. We show that v k = v = v d 2 , i.e., ). We distinguish the following two cases. Then , a contradiction to the choice of G * . Therefore, v = v d+1 , i.e., G * ∼ = ∆ d n . Case 2. d is even. By an argument similar to that of Case 1, we have Proof of Claim 4. Suppose k = l. Then s ≥ d + 2 and k = 0, d. Since d ≥ 4, we can assume k ≥ 2 (otherwise, relabel the vertices in P d ).
By Claims 5 and 3, if l = k + 1, then G is the unicyclic graph with maximum Estrada index of diameter d obtained from P d and v d+1 by adding the edges v k v d+1 and v k+1 v d+1 , and attaching n − d − 2 pendant edges at one vertex v ∈ V(P) ∪ {v d+1 } for some 1 ≤ k ≤ d − 1. By Lemma 10, we get Claim 6. If l = k + 1, then G ∼ = ∆ d n .
By Claims 5 and 3, if l = k + 1, then G is the unicyclic graph with the maximum Estrada index of diameter d obtained from P d and v d+1 by adding the edges v k v d+1 and v k+2 v d+1 , and attaching n − d − 2 pendant edges at one vertex v ∈ V(P) for some 1 ≤ k ≤ d − 2. By Lemma 11, we get Claim 7. If l = k + 2, then G ∼ = G 1 if d is odd, and G ∈ {G 1 , G 2 } if d is even. Now the proof is complete.
By Theorem 3, we can easily obtain the following corollary. Liu et al. in [35] showed the following result on the spectral radii of unicyclic graphs.
Based on Theorems 3 and 4 and previous results on extremal values of Estrada index and spectral radius, we propose the following hypothesis. Hypothesis 1. Let G be a graph in U (n, d). Then EE(G) ≤ EE(∆ d n ), with equality if and only if G ∼ = ∆ d n .

Remark 2.
To prove Hypothesis 1, it suffices to show that EE(∆ d n ) > EE(G 1 ) and EE(∆ d n ) > EE(G 2 ) by Theorem 3. To show this, by previous methods and (1), it suffices to show that for i = 1, 2, the inequality M k (∆ d n ) ≥ M k (G i ) holds for each k ≥ 0 and is strict for some k 0 > 0. However, this can not happen since M 4 (∆ d n ) = 2 Notice that G 1 and G 2 are both bipartite graphs.
The hypothesis is true if we can show that for i = 1, 2, ! holds for all k > 0 and is strict for some k 0 > 0.

Conclusions
In [1], Estrada proposed a graph invariant (the Estrada index) based on a Taylor series expansion of spectral moments. In this paper, we gave some transformations that can be used to compare the Estrada indices of two graphs. As applications, we determined the graphs with the maximum Estrada indices among all unicyclic graphs with fixed diameter d. We showed two candidate extremal graphs if d is odd and three candidate extremal graphs if d is even. For future research, it would be interesting to study Hypothesis 1.