The Minimum Hosoya Index of a Kind of Tetracyclic Graph

Xueji Jiu

School of Mathematics and Statistics, Qinghai Nationalities University, Xining, China.

DOI: 10.4236/jamp.2023.1111215                

Abstract

Let be a graph with n vertices and m edges. The sum of absolute value of all coefficients of matching polynomial is called Hosoya index. In this paper, we determine 2^nd to 4^th minimum Hosoya index of a kind of tetracyclic graph, with m = n +3.

Keywords

Matching Polynomial, Hosoya Index, Tetracyclic Graph, Extremal Graph

1. Introduction

The total number of matchings of agraph is a graphic invariant which is important in structural chemistry. In the chemistry literature this graphic invariant is called the Hosoya index of a molecular graph. It was applied to correlations with boiling points, entropies, calculated bond orders, as well as for coding of chemical structures [1] [2] [3] . Therefore, the ordering of molecular graphs in terms of their Hosoya indices is of interest in chemical thermodynamics. Let $G=\left(V\left(G\right)\mathrm{,}E\left(G\right)\right)$ be a graph with vertex $V\left(G\right)=\left\{v_1\mathrm{,}{v}_2\mathrm{,}\cdots \mathrm{,}{v}_n\right\}$ and edge set $E\left(G\right)=\left\{e_1\mathrm{,}{e}_2\mathrm{,}\cdots \mathrm{,}{e}_n\right\}$ .

The matching polynomial of G is defined as

$\mu \left(G\mathrm{,}x\right)={\displaystyle \underset{k\ge 0}{\sum }}{\left(-1\right)}^{k}m\left(G\mathrm{,}k\right)x^{n-2k},$

where $m\left(G\mathrm{,}k\right)$ the number of its k-matchings. It is convenient to denote $m\left(G\mathrm{,0}\right)=1$ and $m\left(G\mathrm{,}k\right)=0$ for $k>\left[n/2\right]$ . Its theory is well elaborated [4] [5] . The Hosoya index of G, denoted by $Z\left(G\right)$ , is defined as the sum of all the numbers of its matchings, namely

$Z\left(G\right)={\displaystyle \underset{k\ge 0}{\sum }}m\left(G\mathrm{,}k\right)\mathrm{.}$

Let $G_{n\mathrm{,}m}$ be the collection of connected simple graphs of order n and size m. Checking the structure of G in $G_{n\mathrm{,}m}$ , it is easy to see that if $m=n-1,n,n+1,n+2,n+3,\cdots ,n^2-n$ , then G contains at least $m-n+1$ cycles. And these graphs is called unicyclic graphs, bicyclic graphs, tricyclic graphs, tetracyclic graphs, $\cdots$ , respectively. Liu et al. [6] determined the tetracyclic graphs has at least 4 cycles andat most 15 cycles but has no 9 cycles.

The first chemical application of $Z\left(G\right)$ was proposed in 1971 by a chemist Hosoya, which was used to describe the thermodynamic properties of saturated hydrocarbons. Wanger and Gutman [7] gave a summary of the Hosoya index of graphs. Hosoya index conducted important research on the progress of its research. And Wanger [1] proved among all n-vertex, the path P_n has the maximum Hosoya index and the star S_n has the minimum Hosoya index. Ou [8] and [9] studied the unicyclic has the maximum and minimum Hosoya index. Deng [10] and [11] studied the bicyclic has the maximum and minimum Hosoya index. Huang et al. [12] give sharp bounds on the Hosoya index for connected graphs of fixed size. Liu et al. [13] determined the maximum Hosoya index of unicyclic graphs with n vertices and diameter 3 or 4. Their results somewhat answer a question proposed by Wagner and Gutman. In 2010 for unicyclic graphs with small diameter. Liu et al. [14] determined the maximum Hosoya index of tricyclic graphs and the correspond-ing extremal graphs. Li et al. [15] determined the minimum Hosoya index of tricyclic graphs and the corresponding extremal graphs.

In this paper, we are organized as follows. In Section 1, we present some preliminaries and list of some previously known results about Hosoya indices of graphs. In Section 1, we determine the second fourth Hosoya indices of a kind of tetracyclic graph. In final section, we give a brief summary of this paper.

2. Preliminaries

In this section, we introduced some notations and definitions form traditional graph theory, not described here, we refer to [16] . We present some definitions and lemmas to prove the main results later.

Let $G=\left(V\left(G\right)\mathrm{,}E\left(G\right)\right)$ be a simple connected graph with vertex set $V\left(G\right)=\left\{v_1\mathrm{,}{v}_2\mathrm{,}\cdots \mathrm{,}{v}_n\right\}$ and edge set $E\left(G\right)=\left\{e_1\mathrm{,}{e}_2\mathrm{,}\cdots ,e_n\right\}$ . Let $G_{n\mathrm{,}m}$ be the collection of connected simple graphs of order n and size m. Checking the structure of G in $G_{n\mathrm{,}m}$ , it is easy to see that if $m=n-\mathrm{1,}n\mathrm{,}n+\mathrm{1,}n+\mathrm{2,}n+\mathrm{3,}\cdots \mathrm{,}{n}^2-n$ , then G contains at least $m-n+1$ cycles. And these graphs is called unicyclic graphs, bicyclic graphs, tricyclic graphs, tetracyclic graphs, $\cdots$ . If $W\subset V\left(G\right)$ , we denote by $G-W$ the subgraph of G obtained by deleting the vertices of W and the edges incident with them. Similarly, if $E\subset E\left(G\right)$ , we denote by $G-E$ the subgraph of G obtained by deleting the edges of E. If $W=\left\{v\right\}$ and $E=\left\{xy\right\}$ , we write $G-v$ and $G-xy$ instead of $G-\left\{v\right\}$ and $G-\left\{xy\right\}$ , respectively. Denote the neighborhood of $v\in V\left(G\right)$ by $N\left(v\right)=N_G\left(v\right)$ ; and let $N\left[v\right]=N\left(v\right)\cup \left\{v\right\}$ . $d_G\left(v\right)=\left|N_G\left(v\right)\right|$ is vertex v of degree in G. Through out the paper we denote by $P_n\mathrm{,}{C}_n\mathrm{,}{F}_n\mathrm{,}{T}_n\mathrm{,}{S}_n$ the n-vertex graph equals to the path, cycle, forest, tree, star, let $S_n^{+}$ be the graph obtained by two vertex attaching to two pedant vertex in $S_n$ , respectively. For two connected graphs $G_1\mathrm{,}{G}_2$ with $V\left(G_1\right)\cap V\left(G_2\right)=\left\{v\right\}$ , let $G=G_{1}v{G}_2$ be a graph defined by $V\left(G\right)=V\left(G_1\right)\cup V\left(G_2\right)$ , $V\left(G_1\right)\cap V\left(G_2\right)=\left\{v\right\}$ , $E\left(G\right)=E\left(G_1\right)\cup E\left(G_2\right)$ .

In the following we introduce some graph transformation which does not increase the Hosoya index of a graph.

Lemma 2.1. ( [14] ) The Hosoya index of a graph satisfies the following identities:

(i) If $v\in V\left(G\right)$ . Then

$Z\left(G\right)=Z\left(G-v\right)+{\displaystyle \underset{u\in N_G}{\sum }}\left(v\right)Z\left(G-u-v\right).$

(ii) If $uv\in E\left(G\right)$ . Then

$Z\left(G\right)=Z\left(G-uv\right)+Z\left(G-u-v\right)\mathrm{.}$

(iii) If $G_1\mathrm{,}{G}_2\mathrm{,}\cdots \mathrm{,}{G}_k$ are the connected components of a graph G. Then

$Z\left(G\right)={\displaystyle \underset{i=1}{\overset{k}{\prod }}}Z\left(G_i\right).$

Definition 2.1. Suppose that $uv\in E\left(G\right)$ , $N_G\left(u\right)=\left\{v\mathrm{,}{w}_1\mathrm{,}{w}_2\mathrm{,}\cdots \mathrm{,}{w}_s\right\}$ , where $d\left(w_i\right)=1\left(1\le i\le s\right)$ . Let $G^{\mathrm{<mo>\; *\; </mo>}}=G-\left\{u{w}_1\mathrm{,}u{w}_2\mathrm{,}\cdots \mathrm{,}u{w}_s\right\}+\left\{v{w}_1\mathrm{,}v{w}_2\mathrm{,}\cdots \mathrm{,}v{w}_s\right\}$ as shown in Figure 1. We designate the transformation from G to $G_i^{\mathrm{<mo>\; *\; </mo>}}\left(i=\mathrm{1,2}\right)$ in Figure 1 as of type I.

Lemma 2.2. [10] Let G and $G^{\mathrm{<mo>\; *\; </mo>}}$ be two graphs with n vertices defined in Definition 2.1. Then $Z\left(G\right)>Z\left(G^{*}\right)$ .

Definition 2.2. Let H, X and Y be three connected graphs. Suppose that u, v are two vertices of H, v' is a vertex of X, u' is a vertex of Y. Let G be the graph obtained from H, X, Y by identifying v with v' and u with u', respectively. Let $G_1^{\mathrm{<mo>\; *\; </mo>}}$ be the graph obtained from H, X and Y by identifying vertices v, v' and u', and let $G_2^{\mathrm{<mo>\; *\; </mo>}}$ be the graph obtained from H, X and Y by identifying vertices u, v' and u', see Figure 2. We designate the transformation from G to $G_i^{\mathrm{<mo>\; *\; </mo>}}\left(i=\mathrm{1,2}\right)$ in Figure 2 as of type II.

Lemma 2.3. [18] Let $G_1^{\mathrm{<mo>\; *\; </mo>}}$ and $G_2^{\mathrm{<mo>\; *\; </mo>}}$ be three graphs with n vertices defined in Definition 2.2. Then $Z\left(G\right)>Z\left(G_1^{*}\right)$ or $Z\left(G\right)>Z\left(G_2^{*}\right)$ .

Definition 2.3. Let G₀ be a non-trivial connected graph and $u_0\in V\left(G_0\right)$ . Assume that $H\cong C_3$ and $u\mathrm{,}v\in V\left(H\right)$ . Suppose that $G=\left(G_0⊳u_0=u⊲H\right)$ .

Figure 1. Graphs $G\mathrm{,}{G}^{\mathrm{<mo>\; *\; </mo>}}$ .

Suppose that T is a star tree of order n, whose center vertex is w. If $G_1=\left(G⊳u=w⊲T\right)$ , $G_2=G⊳v=w⊲T$ , we designate the transformation from G₁ to G₂ in Figure 3 as of type III.

Lemma 2.4. [17] Let G₁ and G₂ be three graphs with n vertices defined in Definition 2.3. Then $Z\left(G_1\right)<Z\left(G_2\right)$ .

Definition 2.4. Let G be a graph with k vertices, and let $P_k=x_1,x_2,\cdots ,x_k\left(k\ge 3\right)$ be a path in $d_{G\left(x_i\right)}=2\left(i=1,2,\cdots ,k-1\right)$ . Let $G^{\mathrm{<mo>\; *\; </mo>}}$ be a graph of order n is obtained from G by deleting $x_2{x}_3$ and adding $x_1{x}_3$ , see Figure 4. We designate the transformation from G to $G^{\mathrm{<mo>\; *\; </mo>}}$ in Figure 4 as of type IV.

Lemma 2.5. [18] Let G and $G^{\mathrm{<mo>\; *\; </mo>}}$ be two graphs with n vertices defined in Definition 2.4. Then $Z\left(G\right)>Z\left(G^{*}\right)$ .

Lemma 2.6. [19] Let G be a graph, and let $u\mathrm{,}v\in V\left(G\right)$ . Suppose that $G_{s\mathrm{,}t}$ be a graph obtained from G by attaching $s\mathrm{,}t$ pendant vertices to v and u, respectively. Then

$Z\left(G_{s+i,t-i}\right)<Z\left(G_{s,t}\right),1\le i\le t;Z\left(G_{s-i,t+i}\right)<Z\left(G_{s,t}\right),1\le i\le s.$

3. The Minimum Hosoya Index of a Kind of Tetracyclic Graph

Pan [20] determined the minimum Hosoya index among all graphs of n vertices

Figure 2. Graphs $G_1^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}G\mathrm{,}{G}_2^{\mathrm{<mo>\; *\; </mo>}}$ .

Figure 3. Graphs $G_1\mathrm{,}{G}_2$ .

Figure 4. Graphs $G\mathrm{,}{G}^{\mathrm{<mo>\; *\; </mo>}}$ .

and m edges, where $n+3\le m\le 2n-3$ . In the paper, we characterize the 2^nd to 4^th minimum Hosoya index of a kind of tetracyclic Graph, with $m=n+3$ .

The following a kind of tetracyclic graphs and extremal graph defined as follows:

- $p\mathrm{,}q\mathrm{,}m\mathrm{,}l\ge \mathrm{1,}r\ge 0$ , let $F_n^{n+3}\left(p\mathrm{,}q\mathrm{,}r\mathrm{,}m\mathrm{,}l\right)$ be the graph consisting of two given vertices joined by five disjoint paths whose order are $p\mathrm{,}q\mathrm{,}r,m\mathrm{,}l$ , respectively, where $p\mathrm{,}q\mathrm{,}r,m\mathrm{,}l\ge 0$ and most one of them is 0. The resulting graph can be seen in Figure 5.

- $n\ge 7$ , Let $F_{n+3}^{\mathrm{<mo>\; *\; </mo>}}\left(\mathrm{1,1,0,1,1,}n-6\right)$ be the graph obtained by adding $n-6$ pendent vertices to one of two vertices of degree 4 in Figure 5.

Theorem 3.1. [20] Let $G\in F_n^{n+3}$ be a tetracyclic graph with $n\left(n\ge 7\right)$ vertices. $Z\left(G\right)\ge Z\left(F_{n+3}^{*}\left(1,1,0,1,1,n-6\right)\right)=5n-8$ .

We Determine 2^nd to 4^th Minimum Hosoya Index of a Kind of Tetracyclic Graph

By Theorem 3.1 and Lemmas 2.2, 2.3, 2.4 and 2.5, we can obtain a fact as follows. Let $G\in \left\{F_n^{n+3}-F_{n+3}^{\mathrm{<mo>\; *\; </mo>}}\left(\mathrm{1,1,0,1,1,}n-6\right)\right\}$ with n vertices. By repeated applications of transformations I, II, III and IV presented in Definitions 2.1, 2.2, 2.3 and 2.4, respectively. We can transform G into $F_{n+3}^{\mathrm{<mo>\; *\; </mo>}}\left(\mathrm{1,1,0,1,1,}n-6\right)$ . That is, there exist graphs $G^{\left(i\right)}$ for $0\le i\le l$ such that

$G=G^0$ ↪ $G^1$ ↪ $G^2$ ↪ $\cdots$ ↪ $G^{l-1}$ ↪ $G^l=F_{n+3}^{\mathrm{<mo>\; *\; </mo>}}\left(\mathrm{1,1,0,1,1,}n-6\right)\mathrm{,}$ (1)

where $G^{\left(l-1\right)}\ne F_{n+3}^{\mathrm{<mo>\; *\; </mo>}}\left(\mathrm{1,1,0,1,1,}n-6\right)$ . This implies that $G^{\left(l-1\right)}$ has six possible structures, see Figure 6.

Lemma 3.1. Let $A_1\left(s\mathrm{,}t\right)$ be a graph with $n=s+t+6$ vertices. If $n\ge \mathrm{8,}s\ge \mathrm{1,}t\ge 1$ , in $A_1\left(s\mathrm{,}t\right)$ . Then $Z\left(A_1\left(s\mathrm{,}t\right)\right)\ge 6n-15$ , where the equality holds if and only if $A_1\left(s\mathrm{,}t\right)\cong A_1\left(\mathrm{1,}n-7\right)$ .

Proof. By lemma 2.1 and 2.6, we have, $Z\left(A_1\left(s,t\right)\right)\ge Z\left(A_1\left(1,n-7\right)\right)=Z\left(A_1\left(n-7,1\right)\right)=6n-15$ , $Z\left(A_1\left(s,t\right)\right)\ge Z\left(A_1\left(2,n-8\right)\right)=Z\left(A_1\left(n-8,2\right)\right)=7n-24$ . So $Z\left(A_1\left(s,t\right)\right)\ge Z\left(A_1\left(1,n-7\right)\right)=Z\left(A_1\left(n-7,1\right)\right)=6n-15$ .¨

Lemma 3.2. Let $A_2\left(s\mathrm{,}t\right)$ be a graph with $n=s+t+6$ vertices. If

$n\ge \mathrm{8,}s\ge \mathrm{1,}t\ge 1$ , in $A_2\left(s\mathrm{,}t\right)$ . Then $Z\left(A_2\left(s,t\right)\right)\ge 9n-27$ , where the equality

Figure 5. Graphs $F_n^{n+3}\left(p\mathrm{,}q\mathrm{,}r\mathrm{,}m\mathrm{,}l\right)\mathrm{,}{F}_{n+3}^{\mathrm{<mo>\; *\; </mo>}}\left(\mathrm{1,1,0,1,1,}n-6\right)$ .

Figure 6. Graphs $A_1\left(s\mathrm{,}t\right)\mathrm{,}{A}_2\left(s\mathrm{,}t\right)\mathrm{,}{A}_3\left(n-6\right)\mathrm{,}{A}_4\left(n-7\right)\mathrm{,}{A}_5\left(n-7\right)\mathrm{,}{A}_6\left(s\mathrm{,}t\right)$ .

holds if and only if $A_2\left(s,t\right)\cong A_2\left(n-\mathrm{7,1}\right)$ .

Proof. By lemma 2.1 and 2.6, we have, $Z\left(A_2\left(s,t\right)\right)\ge Z\left(A_2\left(n-\mathrm{7,1}\right)\right)=9n-27$ or $Z\left(A_2\left(s,t\right)\right)\ge Z\left(A_2\left(\mathrm{1,}n-7\right)\right)=17n-92$ . So $Z\left(A_2\left(s,t\right)\right)\ge Z\left(A_1\left(1,n-7\right)\right)>Z\left(A_2\left(n-7,1\right)\right)$ .¨

Lemma 3.3. Let $A_6\left(s\mathrm{,}t\right)$ be a graph with $n=s+t+7$ vertices. If $n\ge \mathrm{9,}s\ge \mathrm{0,}t\ge 1$ , in $A_6\left(s\mathrm{,}t\right)$ . Then $Z\left(A_6\left(s,t\right)\right)\ge 17n-33$ , where the equality holds if and only if $A_6\left(s\mathrm{,}t\right)\cong A_6\left(n-\mathrm{8,1}\right)$ .

Proof. By lemma 2.1 and 2.6, we have, $Z\left(A_6\left(s,t\right)\right)\ge Z\left(A_6\left(n-\mathrm{8,1}\right)\right)=17n-33$ or $Z\left(A_6\left(s,t\right)\right)\ge Z\left(A_6\left(\mathrm{0,}n-7\right)\right)=18n-99$ . So $Z\left(A_6\left(s,t\right)\right)\ge Z\left(A_6\left(0,n-7\right)\right)>Z\left(A_6\left(n-8,1\right)\right)$ .¨

Theorem 3.2. Let $G\in F_n^{n+3}$ with $n\ge 7$ vertices. Then $Z\left(G\right)\ge 6n-10>6n-11>6n-15$ , where the equality holds if and only if $G\cong A_1\left(\mathrm{1,}n-7\right)$ .

Proof. By lemma 2.1 and 2.6, we have, $Z\left(A_3\left(n-6\right)\right)=12n-61$ , $Z\left(A_4\left(n-7\right)\right)=6n-11$ and $Z\left(A_5\left(n-7\right)\right)=6n-10$ . Combing Theorem 3.1, Lemmas 3.1, 3.2 and 3.3, (1) and arguments as above, we get that $Z\left(G\right)\ge 6n-10>6n-11>6n-15$ .¨

Now we characterize the extremal graphs with the third minimal Hosoya index in $F_n^{n+3}$ By (1), we know that the extremal graphs with the third minimal Hosoya index will be yielded in $G^{\left(l-1\right)}$ or $G^{\left(l-2\right)}$ . By the reverse operations of I, II, III and IV, we determine the structures of graphs $A_1\left(s,t\right)\mathrm{,}{A}_4\left(n-7\right)$ and $A_5\left(n-7\right)$ in $G^{\left(l-2\right)}$ . And we also determine the lower bounds of Hosoya indices of these graphs.

By the reverse operations of I, II, III and IV, we can obtain that the structures of graphs $A_1\left(s\mathrm{,}t\right)$ in $G^{\left(l-2\right)}$ is isomorphic to one of graphs $A_1^1\left(s\mathrm{,}t\mathrm{,}u\right)$ , $A_1^2\left(s\mathrm{,}t\right)$ $A_1^3\left(s\mathrm{,}t\right)$ , $A_1^4\left(s\mathrm{,}t\right)$ and $A_1^5\left(s\mathrm{,}t\mathrm{,}u\right)$ , see Figure 7.

Lemma 3.4. Let $A_1^1\left(s\mathrm{,}t\mathrm{,}u\right)$ be a graph with $n=s+t+u+6$ vertices. If $n\ge \mathrm{9,}s\ge \mathrm{1,}t\ge 1$ and $u\ge 1$ . Then $Z\left(A_1^1\left(s\mathrm{,}t\mathrm{,}u\right)\right)\ge 11n-40$ , where the equality

Figure 7. Graphs $A_1^1\left(s\mathrm{,}t\mathrm{,}u\right)$ , $A_1^2\left(s\mathrm{,}t\right)$ $A_1^3\left(s\mathrm{,}t\right)$ , $A_1^4\left(s\mathrm{,}t\right)$ , $A_1^5\left(s\mathrm{,}t\mathrm{,}u\right)$ .

holds if and only if $A_1^1\left(s\mathrm{,}t\mathrm{,}u\right)\cong A_1^1\left(\mathrm{1,}n-\mathrm{8,1}\right)$ or $A_1^1\left(n-\mathrm{8,1,1}\right)$ .

Proof. Assume that two of s,t and u are equal to 1 in $A_1^1\left(s\mathrm{,}t\mathrm{,}u\right)$ . By Lemma 2.6, $Z\left(A_1^1\left(\mathrm{1,}n-\mathrm{8,1}\right)\right)=11n-40$ , $Z\left(A_1^1\left(n-\mathrm{8,1,1}\right)\right)=11n-40$ , $Z\left(A_1^1\left(\mathrm{1,1,}n-8\right)\right)=22n-143$ .

Assume that at most one of s,t and u are equal to 1 in $A_1^1\left(s\mathrm{,}t\mathrm{,}u\right)$ . By Lemma 2.6, we get that $\begin{array}{l}Z\left(A_1^1\left(s,t,u\right)\right)=5n+10u+st+4tu+3su+stu-30\\ =s\left(3u-6\right)+\left(4u-6\right)t+\left(st+4\right)u+st-30\end{array}$ . Suppose that $\begin{array}{l}f\left(s,t,u\right)=5n+10u+st+4tu+3su+stu-30-11n+40\\ =s\left(3u-6\right)+\left(4u-6\right)t+\left(st+4\right)u+st+28\end{array}$ . If $s=1,u\ge 2$ and $t\ge 2$ , then $f\left(s,t,u\right)=t\left(5u-5\right)+7u+28>0$ . If $u=1,t\ge 2$ and $s\ge 2$ , then $f\left(s,t,u\right)=s\left(2t-3\right)+2t+32>0$ . If $t=1,s\ge 2$ and $u\ge 2$ , then $f\left(s,t,u\right)=s\left(4u-5\right)+8u+22>0$ . By arguments as above, we have $Z\left(A_1^1\left(s\mathrm{,}t\mathrm{,}u\right)\right)\ge 11n-40$ . where the equality holds if and only if $A_1^1\left(s\mathrm{,}t\mathrm{,}u\right)\cong A_1^1\left(\mathrm{1,}n-\mathrm{8,1}\right)$ or $A_1^1\left(n-\mathrm{8,1,1}\right)$ .¨

Lemma 3.5. Let $A_1^3\left(s\mathrm{,}t\right)$ be a graph with $n=s+t+7$ vertices. If $n\ge \mathrm{8,}s\ge \mathrm{1,}t\ge 1$ , in $A_1^3\left(s\mathrm{,}t\right)$ . Then $Z\left(A_1^3\left(s\mathrm{,}t\right)\right)\ge 10n-40$ , where the equality holds if and only if $A_1^3\left(s\mathrm{,}t\right)\cong A_1^3\left(\mathrm{1,}n-8\right)$ or $A_1^3\left(n-\mathrm{8,1}\right)$ .

Proof. By lemma 2.1 and 2.6, we have, $Z\left(A_1^3\left(s,t\right)\right)\ge Z\left(A_1^3\left(n-8,1\right)\right)=10n-40$ or $Z\left(A_1^3\left(s,t\right)\right)\ge Z\left(A_1^3\left(\mathrm{1,}n-8\right)\right)=10n-40$ . So $Z\left(A_1^3\left(s,t\right)\right)\ge 10n-40$ , where the equality holds if and only if $A_1^3\left(s\mathrm{,}t\right)\cong A_1^3\left(\mathrm{1,}n-8\right)$ or $A_1^3\left(n-\mathrm{8,1}\right)$ .¨

Lemma 3.6. Let $A_1^4\left(s\mathrm{,}t\right)$ be a graph with $n=s+t+7$ vertices. If $n\ge \mathrm{8,}s\ge \mathrm{1,}t\ge 1$ , in $A_1^4\left(s,t\right)$ . Then $Z\left(A_1^4\left(s,t\right)\right)\ge 7n-19$ , where the equality holds if and only if $A_1^4\left(s\mathrm{,}t\right)\cong A_1^4\left(\mathrm{1,}n-8\right)$ or $A_1^4\left(n-\mathrm{8,1}\right)$ .

Proof. By lemma 2.1 and 2.6, we have, $Z\left(A_1^4\left(s,t\right)\right)\ge Z\left(A_1^4\left(n-\mathrm{8,1}\right)\right)=7n-19$ or $Z\left(A_1^4\left(s,t\right)\right)\ge Z\left(A_1^4\left(\mathrm{1,}n-8\right)\right)=7n-19$ . So $Z\left(A_1^4\left(s,t\right)\right)\ge 7n-19$ , where the equality holds if and only if $A_1^4\left(s,t\right)\cong A_1^4\left(\mathrm{1,}n-8\right)$ or $A_1^4\left(n-\mathrm{8,1}\right)$ .¨

Lemma 3.7. Let $A_1^5\left(s\mathrm{,}t\mathrm{,}u\right)$ be a graph with $n=s+t+u+7$ vertices. If $n\ge \mathrm{10,}s\ge \mathrm{0,}t\ge 0$ and $u\ge 0$ . then $Z\left(A_1^5\left(s\mathrm{,}t\mathrm{,}u\right)\right)\ge 9n-27$ , where the equality holds if and only if $A_1^5\left(s\mathrm{,}t\mathrm{,}u\right)\cong A_1^5\left(\mathrm{1,}n-\mathrm{7,0}\right)$ .

Proof. By lemma 2.1 and 2.6, firstly, $s\ge \mathrm{0,}t\ge \mathrm{0,}u\ge 0$ , we need to discuss the following two cases: assume that at most one of s,t and u are equal to 1 in $A_1^5\left(s\mathrm{,}t\mathrm{,}u\right)$ , has three subcases:

(I) If $s=0,t=1$ or $u=1$ ,

(i) $Z\left(A_1^5\left(s\mathrm{,}t\mathrm{,}u\right)\right)\ge Z\left(A_1^5\left(\mathrm{0,1,}n-8\right)\right)=22n-179$ ,

(ii) $Z\left(A_1^5\left(s\mathrm{,}t\mathrm{,}u\right)\right)\ge Z\left(A_1^5\left(\mathrm{0,}n-\mathrm{8,1}\right)\right)=19n-117$ .

(II) If $u=0,t=1$ or $s=1$, has two subcases:

(i) $Z\left(A_1^5\left(s\mathrm{,}t\mathrm{,}u\right)\right)\ge Z\left(A_1^5\left(\mathrm{0,1,}n-7\right)\right)=6n-15$ ,

(ii) $Z\left(A_1^5\left(s\mathrm{,}t\mathrm{,}u\right)\right)\ge Z\left(A_1^5\left(n-\mathrm{8,0,1}\right)\right)=6n-15$ .

(III) If $t=0,s=1$ or $u=1$, has two subcases:

(i) $Z\left(A_1^5\left(s\mathrm{,}t\mathrm{,}u\right)\right)\ge Z\left(A_1^5\left(\mathrm{1,0,}n-8\right)\right)=18n-56$ ,

(ii) $Z\left(A_1^5\left(s\mathrm{,}t\mathrm{,}u\right)\right)\ge Z\left(A_1^5\left(n-\mathrm{8,0,1}\right)\right)=17n-33$ .

Assume that two of s,t and u are equal to 1 in $A_1^5\left(s\mathrm{,}t\mathrm{,}u\right)$ , has three subcases:

(i) $Z\left(A_1^5\left(s\mathrm{,}t\mathrm{,}u\right)\right)\ge Z\left(A_1^5\left(\mathrm{1,1,}n-9\right)\right)=26n-186$ ,

(ii) $Z\left(A_1^5\left(s\mathrm{,}t\mathrm{,}u\right)\right)\ge Z\left(A_1^5\left(\mathrm{1,}n-\mathrm{9,1}\right)\right)=27n-159$ .

(iii) $Z\left(A_1^5\left(s\mathrm{,}t\mathrm{,}u\right)\right)\ge Z\left(A_1^5\left(n-\mathrm{9,1,1}\right)\right)=21n-156$ .¨

Theorem 3.3. Let $G\in F_n^{n+3}$ with $n\ge 7$ vertices. Then $Z\left(G\right)\ge 7n-19>6n-10>6n-11>6n-15$ .

Proof. By lemma 2.1 and 2.6, we have, $Z\left(A_1^2\left(s,t\right)\right)=Z\left(A_2\left(s,t\right)\right)\ge Z\left(A_2\left(1,n-7\right)\right)=9n-27$ , Combing Theorem 3.1, 3.2, Lemmas 3.4, 3.5, 3.6, 3.7, (1) and arguments as above, we get that $Z\left(G\right)\ge 7n-19>6n-10>6n-11>6n-15$ .¨

Similary, by repeated applications of transformations I, II, III and IV, we are considering $A_3\left(n-7\right)$ . This implies that $G^{\left(l-2\right)}$ has six possible structures, see Figure 8.

Lemma 3.8. Let $A_3^2\left(s\mathrm{,}t\right)$ be a graph with $n=s+t+7$ vertices. If

$n\ge \mathrm{9,}s\ge \mathrm{1,}t\ge 1$ , in $A_3^2\left(s\mathrm{,}t\right)$ . Then $Z\left(A_3^2\left(s,t\right)\right)\ge 11n-45$ , where the equality

Figure 8. Graphs $A_3^1\left(s\mathrm{,}t\right)$ , $A_3^2\left(s\mathrm{,}t\right)$ $A_3^3\left(n-7\right)$ , $A_3^4\left(n-8\right)$ , $A_3^5\left(n-8\right)$ , $A_3^6\left(s\mathrm{,}t\right)$ .

holds if and only if $A_3^2\left(s,t\right)\cong A_3^2\left(n-\mathrm{8,1}\right)$ .

Proof. By lemma 2.1 and 2.6, we have, $Z\left(A_3^2\left(s,t\right)\right)\ge Z\left(A_3^2\left(n-\mathrm{8,1}\right)\right)=11n-45$ or $Z\left(A_3^2\left(s,t\right)\right)\ge Z\left(A_3^2\left(1,n-8\right)\right)=14n-75$ . So $Z\left(A_3^2\left(s,t\right)\right)\ge Z\left(A_3^2\left(1,n-8\right)\right)>Z\left(A_3^2\left(n-8,1\right)\right)$ .¨

Lemma 3.9. Let $A_3^6\left(s\mathrm{,}t\right)$ be a graph with $n=s+t+8$ vertices. If $n\ge \mathrm{10,}s\ge \mathrm{0,}t\ge 1$ , in $A_3^6\left(s\mathrm{,}t\right)$ . Then $A_3^6\left(s\mathrm{,}t\right)\ge 9n-30$ , where the equality holds if and only if $A_3^6\left(s\mathrm{,}t\right)\cong A_3^6\left(n-\mathrm{9,1}\right)$ .

Proof. By lemma 2.1 and 2.6, we have, $Z\left(A_3^6\left(s,t\right)\right)\ge Z\left(A_3^6\left(n-\mathrm{9,1}\right)\right)=9n-30$ or $Z\left(A_3^6\left(s,t\right)\right)\ge Z\left(A_3^6\left(\mathrm{0,}n-8\right)\right)=31n-236$ . So $Z\left(A_3^6\left(s,t\right)\right)\ge Z\left(A_3^6\left(0,n-8\right)\right)>Z\left(A_3^6\left(n-9,1\right)\right)$ .¨

Theorem 3.4. Let $G\in F_n^{n+3}$ with $n\ge 11$ vertices. Then $Z\left(G\right)\ge 9n-30>7n-19>6n-10>6n-11>6n-15$ .

Proof. By lemma 2.1 and 2.6, we have, $Z\left(A_3^1\left(s,t\right)\right)=Z\left(A_1^3\left(s\mathrm{,}t\right)\right)\ge 10n-40$ , $Z\left(A_3^3\left(n-7\right)\right)=12n-45$ , $Z\left(A_3^4\left(n-8\right)\right)=14n-53$ , $Z\left(A_3^5\left(n-8\right)\right)=11n-35$ , Combing Theorem 3.1, 3.2, 3.3, Lemmas 3.8, 3.9, (1) and arguments as above, we get that $Z\left(G\right)\ge 9n-30>7n-19>6n-10>6n-11>6n-15$ .¨

Similary, by repeated applications of transformations I, II, III and IV, we are considering $A_4\left(n-7\right)$ . This implies that $G^{\left(l-2\right)}$ has six possible structures, see Figure 9.

Lemma 3.10. Let $A_4^2\left(s\mathrm{,}t\right)$ be a graph with $n=s+t+7$ vertices. If $n\ge \mathrm{9,}s\ge \mathrm{1,}t\ge 1$ , in $A_4^2\left(s\mathrm{,}t\right)$ . Then $Z\left(A_4^2\left(s\mathrm{,}t\right)\right)\ge 8n-14$ , where the equality holds if and only if $A_4^2\left(s\mathrm{,}t\right)\cong A_4^2\left(n-\mathrm{8,1}\right)$ .

Proof. By lemma 2.1 and 2.6, we have, $Z\left(A_4^2\left(s\mathrm{,}t\right)\right)\ge Z\left(A_4^2\left(n-\mathrm{8,1}\right)\right)=8n-14$ or $Z\left(A_4^2\left(s\mathrm{,}t\right)\right)\ge Z\left(A_4^2\left(\mathrm{1,}n-8\right)\right)=11n-52$ . So $Z\left(A_4^2\left(s\mathrm{,}t\right)\right)\ge Z\left(A_4^2\left(\mathrm{1,}n-8\right)\right)>Z\left(A_4^2\left(n-\mathrm{8,1}\right)\right)$ .¨

Lemma 3.11. Let $A_4^6\left(s\mathrm{,}t\right)$ be a graph with $n=s+t+8$ vertices. If

Figure 9. Graphs $A_4^1\left(s\mathrm{,}t\right)$ , $A_4^2\left(s\mathrm{,}t\right)$ $A_4^3\left(n-7\right)$ , $A_4^4\left(n-8\right)$ , $A_4^5\left(n-8\right)$ , $A_4^6\left(s\mathrm{,}t\right)$ .

$n\ge \mathrm{10,}s\ge \mathrm{0,}t\ge 1$ , in $A_4^6\left(s\mathrm{,}t\right)$ . Then $A_4^6\left(s\mathrm{,}t\right)\ge 11n-37$ , where the equality holds if and only if $A_4^6\left(s\mathrm{,}t\right)\cong A_4^6\left(n-\mathrm{9,1}\right)$ .

Proof. By lemma 2.1 and 2.6, we have, $Z\left(A_4^6\left(s\mathrm{,}t\right)\right)\ge Z\left(A_4^6\left(n-\mathrm{9,1}\right)\right)=11n-37$ or $Z\left(A_4^6\left(s\mathrm{,}t\right)\right)\ge Z\left(A_4^6\left(\mathrm{0,}n-8\right)\right)=24n-154$ . So $Z\left(A_4^6\left(s\mathrm{,}t\right)\right)\ge Z\left(A_4^6\left(\mathrm{0,}n-8\right)\right)>Z\left(A_4^6\left(n-\mathrm{9,1}\right)\right)$ .¨

Theorem 3.5. Let $G\in F_n^{n+3}$ with $n\ge 7$ vertices. Then $Z\left(G\right)\ge 9n-30>7n-19>6n-10>6n-11>6n-15$ .

Proof. By lemma 2.1 and 2.6, we have, $Z\left(A_4^1\left(s,t\right)\right)=Z\left(A_1^4\left(s,t\right)\right)\ge 7n-19$ , $Z\left(A_4^3\left(n-7\right)\right)=16n-73$ , $Z\left(A_4^4\left(n-8\right)\right)=10n-31$ , $Z\left(A_4^6\left(n-8\right)\right)=11n-35$ , Combing Theorem 3.1, 3.2, 3.3, 3.4, Lemmas 3.10, 3.11, (1) and arguments as above, we get that $Z\left(G\right)\ge 9n-30>8n-14>7n-19>6n-10>6n-11>6n-15$ .¨

Theorem 3.6. Let $G\in F_n^{n+3}$ with $n\ge 7$ vertices. Then $\begin{array}{l}Z\left(G\right)\ge 6n-10=Z\left(A_4\left(n-7\right)\right)>6n-11=Z\left(A_5\left(n-7\right)\right)>6n-15\\ =Z\left(A_1\left(1,n-7\right)\right)>5n-8=Z\left(F_{n+3}^{*}\left(1,1,0,1,1,n-6\right)\right)\end{array}$ .

4. Conclusions

Combing Theorem 3.1, 3.2, 3.3, 3.4, 3.5 we obtain.

Let $G\in F_n^{n+3}$ with $n\ge 7$ vertices. Then

$\begin{array}{l}Z\left(G\right)\ge 6n-10=Z\left(A_4\left(n-7\right)\right)>6n-11=Z\left(A_5\left(n-7\right)\right)>6n-15\\ =Z\left(A_1\left(1,n-7\right)\right)>5n-8=Z\left(F_{n+3}^{*}\left(1,1,0,1,1,n-6\right)\right)\end{array}$ . In this paper,

we determine the second to fourth minimal Hosoya indices in a kind of tetracyclica graph. This method has not been cited yet, and it is innovative in terms of method. Using this method can solve other graphics and knowledge in the field of graph theory, so promoting the development of graph theory research, respectively. Some new topological indicators in graph theory are closely related to hosoya indicators, laying the foundation for these studies.

Fund Projects

Project No 07M2022003.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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