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The Harary Index of All Unicyclic Graphs with Given Diameter.

1. Introduction

In this paper, all graphs are simple, undirected, and connected. Denote by E([G.sub.1]), V([G.sub.1]) the edge set and vertex set of [G.sub.1] and [absolute value of E([G.sub.1])], [absolute value of V([G.sub.1])] the number of edges and vertices in [G.sub.1], respectively. When [absolute value of E([G.sub.1])] = [absolute value of V([G.sub.1])], the graph [G.sub.1] is called a unicyclic graph. Let [mathematical expression not reproducible] be the degree and the neighborhood of a vertex x in [G.sub.1], respectively. When d(x) = 1, we call x is a pendant vertex. The distance between [x.sub.1] and [y.sub.1] ([x.sub.1], [y.sub.1] [member of] V([G.sub.1])) is denoted by [mathematical expression not reproducible]. The diameter d of [G.sub.1] is [mathematical expression not reproducible]. The graph [G.sub.1] - [x.sub.1][y.sub.1] (or [G.sub.1] + [x.sub.1][y.sub.1]) arisen by deleting the edge [x.sub.1][y.sub.1] [member of] E([G.sub.1]) (or by adding a new edge [x.sub.1][y.sub.1] [not member of] E([G.sub.1])). Let [P.sub.k] and [C.sub.k] be the path and cycle with [absolute value of V([P.sub.k] = [absolute value of V([C.sub.k])] = k, respectively. Denote by P = [v.sub.0][v.sub.1] ... [v.sub.k](k [greater than or equal to] 1) A path of [G.sub.1], where [mathematical expression not reproducible]. When [mathematical expression not reproducible], P is an internal path of [G.sub.1]. When [mathematical expression not reproducible], P is a pendant path of [G.sub.1]. If k = 1, specially, P is a pendant edge. When the subgraph [P.sub.1] is P itself, where [P.sub.1] is induced by V(P) in [G.sub.1], P is an induced path. For other terminologies and notations, we refer the readers to [1].

The Harary index H([G.sub.1]), has been introduced independently in [2, 3]. Its calculation is as follows:

[mathematical expression not reproducible], (1)

where [mathematical expression not reproducible] is defined as above and the sum goes over all the pairs of vertices in [G.sub.1].

The set of n-vertex unicyclic graphs with diameter d is denoted by [U.sub.n,d]. The graph [[DELTA].sup.1.sub.n,d] (see Figure 1) on n vertices arisen from [P.sub.d] = [v.sub.0][v.sub.1] ... [v.sub.d] by attaching n - d - 2 pendant edges to v[??]d/2[??] and adding a new vertex [v.sub.d+1] to be adjacent to [mathematical expression not reproducible].

The graph [[DELTA].sup.2.sub.n,d] (see Figure 1) arisen from [P.sub.d] = [v.sub.0][v.sub.1] ... [v.sub.d] by attaching n - d - 2 pendant edges to [V.sub.[??]d/2[??]+1] and adding a newvertex [v.sub.d+1] to be adjacent to [mathematical expression not reproducible]. The n-vertex graph arisen from [P.sub.d] = [v.sub.0][v.sub.1] ... [v.sub.d] by attaching n - d - 3 pendant edges to [v.sub.[??]d/2][??]and] adding two new adjacent vertices [v.sub.d+1] and [v.sub.d+2] to be adjacent to [v.sub.[??]d/2[??]], denoted by [[DELTA].sub.n,d] (see Figure 2). So, if d = 2, the graph [[DELTA].sub.n,2] arisen from [C.sub.3] by attaching n - 3 pendant edges to one vertex of [C.sub.3].

Note that n-vertex graph [[DELTA].sup.3.sub.n,d] arisen from [P.sub.d] = [v.sub.0][v.sub.1] ... [v.sub.d] by attaching n - d - 3 pendant edges to [v.sub.[??]d/2[??]] and one pendant edge to [v.sub.[??]d/2[??]+1] and adding a newvertex [v.sub.d+1] to be adjacent to [mathematical expression not reproducible]. [[DELTA].sup.4.sub.n,d] arisen from [P.sub.d] = [v.sub.0][v.sub.1] ... [v.sub.d] by attaching n - d- 2 pendant edges to [v.sub.[??]d/2[??]] and adding a new vertex [v.sub.d+1] to be adjacent to [mathematical expression not reproducible].

Let [P.sub.d] = [v.sub.0][v.sub.1] ... [v.sub.d], [U.sub.0]([v.sub.i]) be the graph with d + 2 vertices arisen from [P.sub.d] by adding a new vertex [u.sub.0] to be adjacent to one vertex [v.sub.i]([v.sub.i] [member of] V([P.sub.d]), 1 [less than or equal to] i [less than or equal to] d - 1). [P.sub.d]([C.sub.3]; [v.sub.i]) arisen from [P.sub.d] = [v.sub.0][v.sub.1] ... [v.sub.d] by attaching n - d - 3 pendant vertices to [v.sub.i](1 [less than or equal to] i [less than or equal to] d - 1) and identifying [v.sub.i] with one vertex of [C.sub.3], denote [U.sup.1.sub.n,d] = {[P.sub.d]([C.sub.3]; [v.sub.i]) | 1 [less than or equal to] i [less than or equal to] d -1}. Let [U.sub.d+2] be a (d + 2)-vertex unicyclic graph arisen from [P.sub.d] = [v.sub.0][v.sub.1] ... [v.sub.d] by adding a newvertex [v.sub.d+1] to be adjacent to [v.sub.k] and [v.sub.k+1] (see Figure 2), [U.sub.d+2](n - d -2; [v.sub.j]) arisen from [U.sub.d+2] by attaching n - d - 2 pendant vertices to [v.sub.j](1 [less than or equal to] j [less than or equal to] d - 1, or j = d + 1), and denote [U.sup.2.sub.n,d] = {[U.sub.d+2](n - d - 2; [v.sub.j]) | 1 [less than or equal to] j [less than or equal to] d - 1, or j = d + 1}.

Lately, the investigations on the spectral radius or eigenvalue or topological index of a special class of graphs and the topological properties of a certain class of networks remain a popular topic to researchers. For example, J.Q. Fei and J.H. Tu [4] characterized the complete characterization of the degree Kirchhoff index of bicyclic graphs. J.B. Liu et al. [5] discussed the topological properties of certain neural networks. B. Ning and B. Li [6] discussed the spectral radius of connected claw-free graphs. Y.Y. Wu and Y.J. Chen [7] analyzed the eccentric connectivity index of graphs. C. Wang et al. [8] obtained the least eigenvalue of the graphs whose complements are connected and have pendant paths. Many results can be seen in [2-16] and other papers. In particular, A. Ilic et al. [13] discussed the Harary index among the trees with fixed diameter. In this paper, we determine the graphs which have the maximum and second-maximum Harary indices among all the n-vertex unicyclic graphs with diameter d.

2. Lemmas

Lemma 1 (see [14]). Let A, [H.sub.1], and [H.sub.2] be three connected, pairwise disjoint graphs; note that [u.sub.1], [u.sub.2] [member of] V(A), [v.sub.1] [member of] V([H.sub.1]), [w.sub.1] [member of] V([H.sub.2]). The graph G is obtained by identifying the vertices [u.sub.1], [v.sub.1] and [u.sub.2], [w.sub.1] between A, [H.sub.1], [H.sub.2], respectively. G' is obtained by identifying the vertices [u.sub.1], [v.sub.1], [w.sub.1] from A, [H.sub.1], [H.sub.2] and G" is obtained by identifying the vertices [u.sub.2], [v.sub.1], [w.sub.1] from A, [H.sub.1], [H.sub.2]; then we can get

H(G) < H(G') or H(G) < H(G"). (2)

Lemma 2 (see [9]). Let [G.sub.1] be an n-vertex unicyclic graph and [[DELTA].sub.n,2] be the graph defined as above. Then

H([G.sub.1]) [less than or equal to] H([[nabla].sub.n,2]), (3)

if and only if [G.sub.1] [congruent to] [[nabla].sub.n,2], and the equality holds. When d = 1, then [G.sub.1] [congruent to] [C.sub.3], and when d = 2, n [less than or equal to] 5, then [G.sub.1] [congruent to] [C.sub.4].

Lemma 3. Let [U.sub.0]([v.sub.i]) be the graph defined as above; then

H([U.sub.0]([v.sub.[??]d/2[??]])) [greater than or equal to] H([U.sub.0]([v.sub.i])); (4)

if and only if i = [??]d/2[??], the equality holds.

Proof. By the calculation of Harary index, we can get the following.

[mathematical expression not reproducible]. (5)

[mathematical expression not reproducible]. (6)

When d is odd, we can get [mathematical expression not reproducible].

Thus the result holds.

Lemma 4. Let [U.sup.2.sub.n,d] = [[U.sub.d+2](n - d - 2; [v.sub.j]) | 1 [less than or equal to] j [less than or equal to] d - 1, or j = d + 1} and [[DELTA].sup.1.sub.n,d] and [[DELTA].sup.2.sub.n,d] be the set and two graphs defined as above, respectively. [G.sub.1] [member of] [U.sup.2.sub.n,d]; then

H([G.sub.1]) [less than or equal to] H([[DELTA].sup.1.sub.n,d]); (7)

if and only if [G.sub.1] [congruent to] [[DELTA].sup.1.sub.n,d], the equality holds.

Proof. Choose a graph [G.sub.1] [member of] [U.sup.2.sub.n,d] (3 [less than or equal to] d [less than or equal to] n - 2), such that H([G.sub.1]) is the maximum. The following claims play a crucial role.

Claim 1. [mathematical expression not reproducible].

Proof. First, we prove j [not equal to] d + 1. Otherwise, j = d + 1; denote [mathematical expression not reproducible]; let

[mathematical expression not reproducible]; (8)

using Lemma 1, we get H([G.sub.1]) < H([G.sup.*.sub.1]), a contraction.

So, [mathematical expression not reproducible].

(1) If 1 [less than or equal to] j < k [less than or equal to] [??]d/2[??], let

[G.sup.*.sub.1] = [G.sub.1] - [n-d-2.summation over (i=1)] [v.sub.j][w.sub.i] + [n-d-2.summation over (i=1)] [v.sub.k][w.sub.i]; (9)

if [??]d/2[??] [less than or equal to] k < k + 1 < j [less than or equal to] d - 1, let

[G.sup.*.sub.1] = [G.sub.2] - [n-d-2.summation over (i=1)] [v.sub.j][w.sub.i] + [n-d-2.summation over (i=1)] [v.sub.k+1][w.sub.i]. (10)

(2) If [mathematical expression not reproducible],

let

[G.sup.*.sub.1] = [G.sub.1] - [v.sub.d+1][v.sub.k] - [v.sub.d+1][v.sub.k+1] + [v.sub.d+1][v.sub.j] + [v.sub.d+1][v.sub.j+1]. (11)

(3) If 1 [less than or equal to] j < [??]d/2[??] < k, k + 1 [less than or equal to] d - 1, let

[mathematical expression not reproducible]; (12)

i.e., [G.sup.*.sub.1] [congruent to] [[DELTA].sup.1.sub.n,d];

if 1 [less than or equal to] k < k + 1 < [??]d/2[??] + 1 < j [less than or equal to] d - 1, let

[mathematical expression not reproducible]; (13)

i.e., [G.sup.*.sub.1] [congruent to] [[DELTA].sup.2.sub.n,d].

From Lemma 3 and the calculation of H([G.sub.1]), we can calculate that H([G.sub.1]) < H([G.sup.*.sub.1]), a contraction.

Claim 2. k = [??]d/2[??].

Proof. Otherwise, if k = j < [??]d/2[??], let [G.sup.*.sub.1] = [G.sub.1] - [v.sub.d-1][v.sub.d] + [v.sub.0][v.sub.d]; if k = j > [??]d/2[??], let [G.sup.*.sub.1] = [G.sub.1] - [v.sub.0][v.sub.1] + [v.sub.0][v.sub.d]. From Lemma 3, we can get H([G.sub.1]) < H([G.sup.*.sub.1]), a contraction.

By Claims 1-2, we have the graph [G.sub.1] [congruent to] [[DELTA].sup.1.sub.n,d] or [G.sub.1] [congruent to] [[DELTA].sup.2.sub.n,d].

Claim 3. H([[DELTA].sup.2.sub.n,d]) [less than or equal to] H([[DELTA].sup.1.sub.n,d]).

Proof. If d = 2t is even, we have

[mathematical expression not reproducible]. (14)

If d = 2t + 1 is odd, [[DELTA].sup.1.sub.n,2t+1] [congruent to] [[DELTA].sup.2.sub.n,2t+1, H([[DELTA].sup.1.sub.n,d]) - H([[DELTA].sup.2.sub.n,d]) = 0.

By Claims 1-3, we have [G.sub.1] [congruent to] [[DELTA].sup.1.sub.n,d].

Thus the result follows.

3. Main Results

In this section, we will list our main results.

Theorem 5. Let [U.sub.n,d] and [[DELTA].sup.1.sub.n,d] be defined in Section 1, the n-vertex graph [G.sub.1] [member of] [U.sub.n,d] (3 [less than or equal to] d [less than or equal to] n - 2); then

H([[DELTA].sup.1.sub.n,d]) [greater than or equal to] H([G.sub.1]); (15)

if and only if [G.sub.1] [congruent to] [[DELTA].sup.1.sub.n,d], the equality holds.

Proof. Let [G.sub.1] [member of] [U.sub.n,d]; using Lemma 2, the result holds for d = 1, 2. If d = n - 1, then [G.sub.1] [congruent to] [C.sub.n]. So, we discuss that 3 [less than or equal to] d [less than or equal to] n - 2.

Choose a graph [G.sub.1] [member of] [U.sub.n,d] with H([G.sub.1]) being the maximum. Note that [C.sub.q] is the only cycle and [P.sub.d] = [v.sub.0][v.sub.1] ... [v.sub.d] is the induced path in [G.sub.1]; assume that d([v.sub.0]) = 1; we consider the following claims.

Claim 1. [absolute value of V([C.sub.q]) [intersection] V([P.sup.d])] > 0.

Proof. Otherwise, suppose that there exists a path [P.sub.k] = [v.sub.h][v.sub.g+1] ... [v.sub.g+k-1][v.sub.l] connecting the cycle [C.sub.q] and the path [P.sub.d] with [v.sub.h] [member of] V([P.sub.d]), [v.sub.l] [member of] V([C.sub.q]); denote [mathematical expression not reproducible]. Let

[mathematical expression not reproducible]. (16)

Then, applying Lemma 1, H([G.sub.1]) < H([G.sup.*.sub.1]), a contraction.

Using Claim 1, we note that [mathematical expression not reproducible].

Claim 2. For any v [member of] V([G.sub.1]) \ (V([P.sub.d]) [universal] V([C.sub.q])), d(v) = 1.

Proof. Otherwise, assume that there exists a vertex [mathematical expression not reproducible]. Suppose that [mathematical expression not reproducible]. Let

[mathematical expression not reproducible]. (17)

Then, using Lemma 1, H([G.sub.1]) < H([G.sup.*.sub.1]), a contraction.

By Claim 2, we have any vertex v [member of] V([G.sub.1]) (V([P.sub.d]) [universal] V([C.sub.q])) is pendant vertex.

Claim 3. [absolute value of V([C.sub.q]) [intersection] V([P.sub.d])] [less than or equal to] 2.

Proof. Otherwise, suppose that V([C.sub.q]) [intersection] V([P.sub.d]) = {[v.sub.t], [v.sub.t+1], ..., [v.sub.t+k]}, k [greater than or equal to] 2.

(1) If t < [??]d/2[??] - 1, denote [mathematical expression not reproducible]. Let

[G.sup.*.sub.1] = [G.sub.1] - [v.sub.s][v.sub.t] + [v.sub.s][v.sub.t+1] - [d([v.sub.t])-3.summation over (i=1)][v.sub.t][u.sub.i] + [d([v.sup.t])-3.summation over (i=1)][v.sub.t+1][u.sub.i]. (18)

(2) If [mathematical expression not reproducible]. Let

[mathematical expression not reproducible]. (19)

Then, applying Lemma 3 and the calculation of H([G.sub.1]), H([G.sub.1]) < H([G.sup.*.sub.1]), a contraction.

Claim 4. (1) if [absolute value of V([C.sub.q]) [intersection] V([P.sub.d])] = 2, s = d + 1; (2) if [absolute value of V([C.sub.q]) [intersection] V([P.sub.d])] = 1, s = d + 2.

Proof. (1) Otherwise, we assume that s [greater than or equal to] d + 2. The graph [G.sup.*.sub.1] arisen by deleting all edges in E([C.sub.q]) of [G.sub.1] and incidence with [C.sub.q] except the edges [v.sub.t][v.sub.t+1], [v.sub.t+1][v.sub.d+1] of [G.sub.1], adding the edge [v.sub.t][v.sub.d+1], connecting all isolated vertices to the vertex [v.sub.t] (if t [greater than or equal to] [??]d/2[??]) or to the vertex [v.sub.t+1](if t + 1 [less than or equal to] [??]d/2[??]) of [G.sub.1]. Together with the definition of the Harary index and Lemma 3, H([G.sub.1]) < H([G.sup.*.sub.1]), a contraction.

(2) Otherwise, we assume that s [greater than or equal to] d + 3. The graph [G.sup.*.sub.1] is obtained from [G.sub.1] by deleting all edges in E([C.sub.q]) and incidence with [C.sub.q] except the edges [v.sub.t][v.sub.d+1], [v.sub.d+1][v.sub.d+2], adding the edge [v.sub.t][v.sub.d+2], connecting all isolated vertices to the vertex [v.sub.t]. We can calculate that H([G.sub.1]) < H([G.sup.*.sub.1]), a contraction.

By Claim 4 and Lemmas 1 and 3, we get [G.sub.1] [member of] [U.sup.1.sub.n,d] [universal] [U.sup.2.sub.n,d]. If [G.sub.1] [member of] [U.sup.1.sub.n,d], applying Lemma 3, [G.sub.1] [congruent to] [[nabla].sub.n,d] (see Figure 2); if [G.sub.1] [member of] [U.sup.2.sub.n,d] applying Lemma 4 [G.sub.1] [congruent to] [[DELTA].sup.1.sub.n,d].

Claim 5. H([[DELTA].sup.1.sub.n,d]) > H([[nabla].sub.n,d]).

Proof. If d = 2t is even, we have

[mathematical expression not reproducible]. (20)

If d = 2t + 1 is odd, we have

[mathematical expression not reproducible]. (21)

By Claims 1-5, we have [G.sub.1] [congruent to] [[DELTA].sup.1.sub.n,d].

Thus the result follows.

Theorem 6. Let [U.sub.n,d], [[DELTA].sup.1.sub.n,d] be defined in Section 1; the graph [G.sub.1] [member of] [U.sub.n,d] {[[DELTA].sup.1.sub.n,d]} (3 [less than or equal to] d [less than or equal to] n - 2); then

(1) If d = 2t is even, we obtain

H ([[nabla].sub.n,d]) [greater than or equal to] H([G.sub.1]), when

2n [greater than or equal to] [t.sup.2] + 5t + 2; (22)

if and only if [G.sub.1] [congruent to] [[nabla].sub.n,d], the equality holds.

H([[DELTA].sup.2.sub.n,d]) [greater than or equal to] H([G.sub.1]), when 4t + 8 [less than or equal to] 2n [less than or equal to] [t.sup.2] + 5t + 2; (23)

if and only if [G.sub.1] [congruent to] [[DELTA].sup.2.sub.n,d], the equality holds.

(2) If d = 2t + 1 is odd, we have

H([[DELTA].sup.4.sub.n,d]) [greater than or equal to] H([G.sub.1]); (24)

if and only if [G.sub.1] [congruent to] [[DELTA].sup.4.sub.n,d], the equality holds.

Proof. Choose a graph [G.sub.1] [member of] [U.sub.n,d] {[[DELTA].sup.1.sub.n,d]}, such that H([G.sub.1]) is as large as possible; together with Lemmas 1, 3, and 4 and Theorem 5, we have the following.

(1) If d = 2t is even, [G.sub.1] [congruent to] [U[DELTA].sup.2.sub.n,d] or [G.sub.1] [congruent to] [ [DELTA].sup.2.sub.n,d] or [G.sub.1] [congruent to] [[nabla].sub.n,d].

In fact,

[mathematical expression not reproducible]. (25)

(1.1) When 2n [greater than or equal to] [t.sup.2] + 5t + 2, H([[DELTA].sub.n,d]) - H([[DELTA].sup.2.sub.n,d]) [greater than or equal to] 0.

(1.2) When 4t + 8 [less than or equal to] 2n [less than or equal to] [t.sup.2] + 5t + 2, H([V.sub.n,d]) - H([[DELTA].sup.2.sub.n,d]) [less than or equal to] 0.

(2) If [mathematical expression not reproducible].

[mathematical expression not reproducible]. (26)

Thus the result follows.

https://doi.org/10.1155/2018/3957023

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This paper is supported by the Natural Science Foundation of China (11871077), the National Natural Science Foundation of China (11371028), the Natural Science Foundation of Anhui Province of Anhui (1808085MA04).

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Bao-Hua Xing, (1) Gui-Dong Yu (iD), (1,2) Li-XiangWang, (1) and Jinde Cao (iD) (3)

(1) School of Mathematics and Computation Sciences, Anqing Normal University, Anqing 246133, China

(2) Basic Department, Hefei Preschool Education College, Hefei 230013, China

(3) School of Mathematics, Southeast University, Nanjing210096, China

Correspondence should be addressed to Jinde Cao; jdcao@seu.edu.cn

Received 15 June 2018; Revised 23 July 2018; Accepted 7 August 2018; Published 16 September 2018

Academic Editor: Beatrice Di Bella

Caption: Figure 1: The graphs [[DELTA].sup.1.sub.n,d] and [[DELTA].sup.2.sub.n,d].

Caption: Figure 2: The graphs [[nabla].sub.n,d] and [U.sub.d+2].
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