The extremal pentagon-chain polymers with respect to permanental sum

The permanental sum of a graph G can be defined as the sum of absolute value of coefficients of permanental polynomial of G. It is closely related to stability of structure of a graph, and its computing complexity is #P-complete. Pentagon-chain polymers is an important type of organic polymers. In this paper, we determine the upper and lower bounds of permanental sum of pentagon-chain polymers, and the corresponding pentagon-chain polymers are also determined.

where S k (G) is the collection of all linear subgraphs H of order k in G, and c(H) is the number of cycles in H. Recall that A linear subgraph of a graph G is termed as a subgraph whose components are cycles or single edges.
The permanental sum of G, denoted by PS(G), is the sum of the absolute values of all coefficients of π(G, x) , i.e., Background. The study of permanental polynomial of a graph in chemical literature were started by Kasum et al. 1 . They computed respectively permanental polynomials of paths and cycles, and zeroes of these polynomials. Cash 3 investigated permanental polynomials of some chemical graphs(including benzene, o-biphenylene, coronene, C 20 fullerene). And he pointed out that studying the absolute values of coefficients of permanental polynomials is of interest. However, it is difficult to compute the coefficients of permanental polynomial of a graph. Up to now, only a few about the coefficients of permanental polynomials of chemical graphs and its potential applications seems to have been published [4][5][6][7][8][9][10][11][12][13][14] . A natural problem is researching the sum of coefficients of permanental polynomials of a chemical graph, i.e., how characterize the permanental sum of a chemical graph. There exists a peculiar chemical phenomenon which closely relate to the permanental sum. For the theo-   16 computed the permanental sums of all 271 fullerenes in C 50 . They found that the permanental sum of C 50 (D 5h ) achieves the minimum among all 271 fullerenes in C 50 , and they also pointed out that the permanental sum could be closely related to the stability of molecular graphs. A bad news is the computing complexity of permanental sum is #P-complete 17 . In spite of this difficulty, the studies of permanental sums have received a lot of attention from researchers in recent years. Chou et al. 18 studied the property of C 70 . Li et al. 19 determined the extremal hexagonal chains with respect to permanental sum. Li and Wei 20 characterized the extremal octagonal chains with respect to permanental sum. Wu and Lai 21 study some basic properties of the permanental sum of general graphs, in particular, they pointed out that the permanental sum is closed to the Fibonacci numbers. For the background and some known results about this problem, we refer the reader to [22][23][24][25] and the references therein. In addition, the permanental sum is similar to Hosoya index proposed by Haruo Hosoya. Hosoya index of a graph G, denoted by Z(G), is defined as the total number of independent edge sets of G 26 . The Hosoya index is closely related to the boiling points of chemical graphs. Wu and Lai 21 shown that PS(G) ≥ Z(G) with the equality holds if and only if G is a forest. These indicate that the permanental sum is likely to explain certain characteristics of chemical molecules.
Base on arguments as above, it is interesting to study the permanental sums of chemical graphs.
The graph model of a type of organic polymers. Organic polymers are a fascinating class of chemical materials with a strikingly wide range of applications [27][28][29][30][31][32] . Many of them contain chains of five-membered rings as a building block, see Figure 1 in 33 . It is easy to see that the graph model of the organic polymer with n five-membered rings is an edge-pentagon-chain. An edge-pentagon-chain EPC n with n pentagons, which is a sub-chain of an edge-pentagon-chain, can be regarded as an edge-pentagon-chain EPC n−1 with n − 1 pentagons adjoining to a new terminal pentagon by a cut edge, see Fig. 1. By contracting operation of graphs, an edgepentagon-chain EPC n with n pentagons is changed new pentagon-chain called vertex-pentagon-chain. That is, A vertex-pentagon-chain, denoted by VPC n , is obtained by contracting every cut edge in EPC n , see Fig. 1. Checking the structure of a vertex-pentagon-chain, it is not difficult to see that the vertex-pentagon-chain also is a graph model of a type of organic polymers 34,35 .
In this paper, we focus on properties of permanental sum of pentagon-chain polymers. We hope that results of the paper will provide theoretical support for the study of organic polymers.
Preliminaries. Let EPC n = S 1 S 2 · · · S n be a polyomino chain with n(≥ 2) pentagons, where S k is the k-th pentagon in EPC n attached to S k−1 by a cut edge u k−1 w k , k = 2, 3, . . . , n , where w k = v 1 is a vertex of S k . A vertex v is said to be ortho-and meta-vertex of S k if the distance between v and w k is 1 and 2, denoted by o k and m k , respectively. Checking Fig. 1, it is easy to see that w n = v 1 , ortho-vertices o n = v 2 , v 5 , and meta-vertex An edge-pentagon-chain EPC n is an edge-ortho-pentagon-chain if u k = o k for 2 ≤ k ≤ n − 1 , denoted by EPC o n . An edge-pentagon-chain EPC n is an edge-meta-pentagon-chain if u k = m k for 2 ≤ k ≤ n − 1 , denoted by EPC m n . The resulting graphs see Fig. 2. Contracting every cut edge in EPC o n and EPC m n , the resulting graphs are called a vertex-ortho-pentagon-chain VPC o n and a vertex-meta-pentagon-chain VPC m n , respectively. See Fig. 3. In 21 , some properties of permanental sum of a graph are determined. Lemma 1.1 21 Let P n be a path with n vertices. Then  By Lemma 1.2, we obtain the following corollary.

Results
The bound of permanental sum of edge-pentagon-chains. In order to prove the lemma 2.1, we give two auxiliary graphs. One is denoted by EPC o ′ n obtained from EPC o n deleting a ortho-vertex in S n . The other is denoted by EPC m ′ n obtained from EPC m n deleting meta-vertex in S n . The resulting graphs see Fig. 4.

Lemma 2.1 Let EPC o n and EPC m n be an edge-ortho-pentagon-chain and an edge-meta-pentagon-chain, respectively. Then
Proof By Lemma 1.2, we have Thus, . According to the property of a similarity matrix, we have Therefore, By (1) and (2) (2) .   (3) and (4), we have

The bound of permanental sum of vertex-pentagon-chains.
We first present two auxiliary graphs.
One is denoted by VPC o ′ n obtained from VPC o n deleting a ortho-vertex in S n . The other is denoted by VPC m ′ n obtained from VPC m n deleting meta-vertex in S n . The resulting graphs see Fig. 6. .

PS(VPC
.    VPC n−i such that PS(G ′ ) < PS(G) , which contradicts the hypothesis G attains the minimum permanental sum. Thus, G = VPC m n . Similarly, let G = S 1 S 2 . . . S n ∈ G n be the vertex-pentagon-chain with the largest permanental sum. The following we prove that G = VPC o n . Suppose to the contrary that G = VPC o n . Then there must exist i ∈ (1, 2, . . . , n) such that G = VPC i m VPC n−i . By Theorem 2.1, there exists G ′ = VPC i o VPC n−i such that PS(G ′ ) > PS(G) , which contradicts the hypothesis G attains the maximum permanental sum. Thus, G = VPC o n . By Lemma 2.2 and argument as above, it is straightforward to obtain Theorem 2.4.

Discussions
Determining extremal value is an important problem in scientific research. In this paper, we characterize the tight bound of permanental sums of all edge-pentagon-chains and vertex-pentagon-chains, respectively. And the corresponding graphs are also determined. For an edge-pentagon-chain(resp. vertex-pentagon-chain), using the computing method in Lemma 2.1(resp. Lemma 2.2) can compute the permanental sum of any edge-pentagonchain(resp. vertex-pentagon-chain). For every organic polymers, we always find a graph model corresponding it. Thus, the permanental sum of a organic polymers can be computed by the formulas in Lemma 1.2.
C 50 (D 5h ) is captured and its permanental sum achieves the minimum among all C 50 . Is the phenomenon a coincidence? Does the phenomenon exist for other chemical molecular? These are very interesting problems. However, we cannot answer them. Our motivation is to determine the extremal graphs with respect to permanental sum for some type chemical graphs in this paper. In the future, we will find the answers of the problem as above.