Extremal octagonal chains with respect to the coefficients sum of the permanental polynomial

Abstract

Tree-like octagonal systems are cata-condensed systems of octagons, which represent a class of polycyclic conjugated hydrocarbons. An octagonal chain is a cata-condensed octagonal system with no branchings. In this paper, the extremal octagonal chains with n octagons having the minimum and maximum coefficients sum of the permanental polynomial are identified, respectively.

Introduction

In this paper, we consider simple and finite graphs only and assume that all graphs are connected. All the notations and terminologies not defined here we refer the reader to Bondy and Murty [3].

Let $G=(V_G,E_G)$ be a graph with vertex set V_G and edge set E_G. Then $G-v,$ $G-uv$ denote the graph obtained from G by deleting vertex v ∈ V_G, or edge uv ∈ E_G, respectively (this notation is naturally extended if more than one vertex or edge is deleted). Similarly, $G+uv$ is obtained from G by adding the edge $uv\notin E_G$. Denote by P_n and C_n the path and cycle on n vertices, respectively. Let G₁ and G₂ be two vertex disjoint graphs with $u_1{v}_1\in E_{G_1}$ and $u_2{v}_2\in E_{G_2},$ then u₁v₁ ≡ u₂v₂ if u₁ identifies u₂, v₁ identifies v₂ and multiple edges u₁v₁, u₂v₂ are replaced by one edge.

Let $M=\left(m_{ij}\right)$ be an n × n matrix. The permanent [1] of M is defined as

$$\text{per}M=\sum _{\sigma }{m}_{1{\sigma }_1}{m}_{2{\sigma }_2}\dots m_{n{\sigma }_n},$$

where the sum ranges over all the permutations σ of $\{1,2,\dots ,n\}$. Let $V_G=\{v_1,v_2,\dots ,v_n\}$ and $A\left(G\right)={\left(a_{ij}\right)}_{n\times n}$ be the adjacency matrix of order n whose entries $a_{ij}=1$ if v_i is adjacent to v_j and $a_{ij}=0$ otherwise. The characteristic polynomial of G is

$$\varphi (G,x)=\det (x{I}_n-A\left(G\right)),$$

where I_n is an identity matrix of order n. The permanental polynomial of G is defined as (see [8])

$$\pi (G,x)=\text{per}(xI-A(G\left)\right).$$

(1.1) and (1.2) may be denoted in the coefficients forms, respectively, as

$$\begin{array}{ccc}& & \varphi (G,x)=\det (xI-A\left(G\right))=\sum _{i=0}^n{a}_i{x}^{n-i},\hfill \end{array}$$

$$\begin{array}{ccc}& & \pi (G,x)=\text{per}(xI-A\left(G\right))=\sum _{i=0}^n{b}_i{x}^{n-i}.\hfill \end{array}$$

If G is bipartite, in view of [1], [4], [5], [12], [25], we know that

$${(-1)}^i{a}_{2i}⩾0,b_{2i}⩾0,a_{2i+1}=b_{2i+1}=0\text{for}\text{all}i⩾0$$

and $b_{2i}={\sum }_H\text{per}A\left(H\right),$ where A(H) is the adjacency matrix of the induced subgraph H of G with 2i vertices and the sum ranges over all induced subgraphs H of G with 2i vertices. Furthermore, by [5], [25] we get that the coefficients of the permanental polynomial in (1.4) satisfying the property:

$${(-1)}^i{b}_i=\sum _B{2}^{c\left(B\right)},1⩽i⩽n,$$

where B is an elementary subgraph of G on i vertices (a subgraph B is called an elementary subgraph if all components of B are edges or cycles) and c(B) denotes the number of cycles and the sum ranges over all elementary subgraphs B on i vertices. In particular, if G is bipartite, then one has

$$b_{2i}=\sum _H{\left(m\left(H\right)\right)}^2,1⩽i⩽\lfloor n/2\rfloor ,$$

where m(H) is the number of perfect matchings of H and the sum ranges over all the induced 2i-vertex subgraphs H of G (see [24]).

It is known [29] that the complexity of computation of the permanental polynomials of graphs is NP-complete. Hence, many researchers focused on finding methods to compute the permanental polynomial of some type of graphs; see [2], [6], [20], [22], [25], [30], [31]. It is also interesting to characterize the relationship between the characteristic and permanental polynomials of some chemical graphs. For more details one may be referred to those in [1], [5], [11], [15], [21], [28]. On the other hand, the (signless) Laplacian permanent of graphs was studied extensively; see [7], [9], [13], [14], [17], [19].

In mathematical literature, the coefficients of graph polynomial (for example, Tutte polynomial, Laplacian characteristic polynomial, independence polynomial, etc.) attract more and more researchers’ attention. For more details, one may be referred to those in [16], [18], [26], [27]. Li et al. [23] first studied the coefficients sum of permanental polynomial of hexagonal chains. It is natural and interesting to do further research along this line.

In order to formulate our main results, we need to introduce some notations. An octagonal system is a 2-connected graph consisting with some regular octagons of unit edge length. It seems that the first study on octagonal system in mathematical chemistry is [10]. An octagonal chain is an octagonal system if it has no vertex belonging to three octagons and no octagon with more than two adjacent octagons.

Denote the set of octagonal chains with n octagons by ${\mathcal{G}}_n$. For each octagonal chain G_n in ${\mathcal{G}}_n,$ we may write it as $O_1{O}_2\dots O_n,$ where O_i is the ith octagon in G_n. That is to say, the octagonal chain G_n can be obtained from an octagon by adding octagons gradually for n ≥ 2. Therefore, any octagonal chain $G_n\in {\mathcal{G}}_n$ can be obtained from one $G_{n-1}$ in ${\mathcal{G}}_{n-1}$ by attaching an octagon in the following five cases as depicted in Fig. 1:   (1)  rs ≡ ab; (2)  rs ≡ bc; (3)  rs ≡ cd; (4)  rs ≡ de; (5)  rs ≡ ef, which are denoted by α-type, β-type, γ-type, δ-type, ε-type, respectively. Denote by [T]_k the octagonal chain obtained from an octagonal chain T by attaching an octagon O through k-type attaching, where k ∈ {α, β, γ, δ, ε}. Then each octagonal chain G_n can be written as $G_n=k_1{k}_2\dots k_n,$ where k_i ∈ {α, β, γ, δ, ε} for 1 ≤ i ≤ n.

As it is irrelevant to which type the first and second octagons are, we set $k_1=k_2=\gamma$. Then $G_n=\gamma \gamma k_3\cdots k_n$. If $k_i=\gamma$ for each i, then G_n is a linear chain, denoted by L_n; if k_i ∈ {α, ε} (or {β, δ}) and $k_i\ne k_{i+1}$ for each i ≥ 3, then G_n is called a zigzag chain, denoted by $Z_n^1$ (or $Z_n^2$); if $k_i=\alpha$ (or ε) for each i ≥ 3, then G_n is a helix chain, denoted by $H_n^1$; if $k_i=\beta$ (or δ) for each i ≥ 3, then G_n is also a helix chain, denoted by $H_n^2$ (see Fig. 2). Thus we can see that $G_1=L_1=Z_1^1=Z_1^2=H_1^1=H_1^2,G_2=L_2=Z_2^1=Z_2^2=H_2^1=H_2^2$ and $G_3=\{L_3,Z_3^1=H_3^1,Z_3^2=H_3^2\}$.

In this paper, inspired by the idea of Li et al. [23] to identify the extremal hexagonal chains w.r.t. the coefficients sum of the permanental polynomial, we consider the extremal problem on the coefficients sum of the permanental polynomial of octagonal chains. We will prove the following results.

Theorem 1.1

The helix chain $H_n^2$ uniquely minimizes the coefficients sum of the permanental polynomial among all octagonal chains with n octagons.

Theorem 1.2

The zigzag chain $Z_n^1$ uniquely maximizes the coefficients sum of the permanental polynomial among all octagonal chains with n octagons.

The organization of this paper is as follows. In Section 2, we introduce some auxiliary results on the permanental polynomial, which will be used to study the coefficients sum of some octagonal chains. In Section 3, we first introduce a roll-attaching operation of the octagonal chains. Then we establish some technical lemmas that help us characterize the extremal graphs. Based on the results in the previous sections, we give the proofs of our main results in Section 4. Some concluding remarks are given in the last section.