Abstract
Let be a graph and the adjacency matrix of . The permanent of matrix is called the permanental polynomial of . The permanental sum of is the sum of the absolute values of the coefficients of permanental polynomial of . Computing the permanental sum is #p-complete. In this note, we prove the maximum value and the minimum value of permanental sum of quasi-tree graphs. And the corresponding extremal graphs are also determined. Furthermore,we also determine the graphs with the minimum permanental sum among quasi-tree graphs of order and size , where .
1. Introduction
The permanent of matrix is defined aswhere the sum is taken over all permutations of .
Let be a graph with vertices and let be its adjacency matrix. The permanental polynomial of is defined aswhere is the unit matrix of order . Basic theory of permanental polynomials is well studied recently in [1–3] and the references therein. Kasum et al.[4] and Merris et al. [5] gave the coefficients of the permanental polynomial of , i.e.,where the sum is taken over all Sachs subgraphs of on vertices and is the number of cycles in . Recall that a Sachs subgraph is a graph in which each component is a single edge or a cycle.
The permanental sum of graph , denoted by , can be defined as the summation of all absolute values of coefficients of permanental polynomial of , i.e.,Thus, if is an empty graph. Wu and So [6] have shown that computing permanental sum of a graph is #P-complete.
The permanental sum of a graph was first considered by Tong [7]. In [8], Xie et al. captured a labile fullerene . Tong computed all 271 fullerenes in . In his study, Tong found that the permanental sum of achieves the minimum among all 271 fullerenes in . He pointed that the permanental sum would be closely related to stability of molecular graphs. Recently, the permanental sum of a graph has received much attention. Li et al. [9] determined the extremal hexagonal chains with respect to permanental sum. Li and Wei [10] proved the lower and upper bounds for the permanental sum of an octagonal chain. Wu and Lai [11] systematically introduced the properties of permanental sum of a graph.
A connected graph is called a quasi-tree graph, if there exists a vertex in such that is a tree. Let be a quasi-tree graph with vertices and edges. Then , and the degree of in equals . Denote , and . As an important class of graphs, quasi-tree graphs have been widely studied. For the background and some known results about quasi-tree graphs, we refer the reader to [12–15].
The purpose of this note is to investigate the properties of permanental sum of quasi-tree graphs. The note is organized as follows. In the next section, we review some previous results that will be needed in the sequel. In Section 3, we discuss the permanental sum of quasi-tree graphs.
2. Some Preliminary
In this note, we only consider finite, undirected, and simple graph. Let be a graph with vertex set and edge set . The neighborhood of vertex , denoted by , is the set of vertices adjacent to . The graph that arises from by deleting a vertex or an edge will be denoted by or . Let denote the union of two vertex disjoint graphs and . For any positive integer , denotes the union of disjoint copies of . The path, cycle, and star of order are denoted by , and , respectively.
Two edges of are said to be independent if they are not adjacent in . A -matching of is a set of mutually independent edges. For an integer , let denote the number of -matchings of a graph . The Hosoya index of a graph is defined to be the total number of matchings of , that is,where is the number of the vertices of the graph . Some results on Hosoya indices were studied in [3, 16–18].
For , let denote the sequence of Fibonacci numbers, in particular, and .
Lemma 1 (see [10]). Let be a forest with order and , where is a tree with vertices, . Then with equality if and only if . Moreover , where with equality if and only if .
Let be the empty graph of order . Denote the graph joint of two graphs, and the disjoint union of two graphs. The graphs and are shown in Figure 1[19].
Lemma 2 (see [19]). Let be the set consisting of all graphs of order and size . For with ,Equality holds if and only if , or when , where Graphs and see Figure 1.
By the definitions of and , we obtain that . By Lemma 2, we have the following.
Corollary 3. Let be a quasi-tree graph with . Thenwhere the equality holds if and only if .
Lemma 4 (see [11]). Letting be a tree with order , then , the first equality holds if and only if , and the second equality holds if and only if .
Lemma 5 (see [11]). Let be a path with vertices. Then
Lemma 6 (see [11]). The permanental sum of a graph satisfies the following identities:(i)Let and be two connected graphs. Then(ii)Let be an edge of a graph and the set of cycles containing . Then(iii)Let be a vertex of a graph and the set of cycles containing . Then
By Lemma 6, we have the following.
Corollary 7. Let be a graph and an edge of . Then .
3. Main Results
In this section, we will investigate the properties of permanental sum of a quasi-tree graph.
Theorem 8. Let be a quasi-tree graph with . Thenwhere the equality holds if and only if .
Proof. By the definition of permanental sum of a graph, it can be known that , where denotes the number of all Sachs graphs containing cycles of . Checking , we know that has exactly cycles and . Thus . By Corollary 3, we have with equality if and only if . By Lemma 6, we obtain thatThis completes the proof.
Theorem 9. Let . Thenwhere the equality holds if and only if .
Proof. Let , and let and . Suppose that has the maximum permanental sum. We will characterize the structure of . By (iii) of Lemma 6, it can be known that if has the maximum permanental sum, then , , and must attain maximum value. From the definition of a quasi-tree graph, we know that is a tree. By Lemma 4, has the maximum permanental sum when is isomorphic to path . Since a path has exactly two vertices of degree 1. Thus there exist exactly two vertices, say and , such that and are paths. Set . By Lemma 1, only when has two components, each of which is a path, does attain the maximum value. Similarly, by Lemma 1 and Lemma 6, attains the maximum value if and only if has two components, each of which is a path, and has the largest number of cycles in . Combining arguments above and Corollary 7, must be isomorphic to (see Figure 2). Let the number of -cycles, -cycles,..., -cycles in be , respectively. By Lemma 6, we obtain that
By Theorems 8 and 9, we obtain the following result.
Theorem 10. Let . ThenThe first equality holds if and only if , and the second equality holds if and only if .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research is supported by the National Natural Science Foundation of China (Nos. 11761056, 11801061); the Ministry of Education Chunhui Project (No. Z2017047) and the NSF of Qinghai Province (No. 2016-ZJ-947Q); and the Key Project of QHMU (No. 2019XJZ10).