Abstract
Let be a graph with vertices, and let and denote the Laplacian matrix and signless Laplacian matrix, respectively. The Laplacian (respectively, signless Laplacian) permanental polynomial of is defined as the permanent of the characteristic matrix of (respectively, ). In this paper, we show that almost complete graphs are determined by their (signless) Laplacian permanental polynomials.
1. Introduction
We use to denote a simple graph with vertex set and edge set . The degree of a vertex is denoted by . The degree matrix of , denoted by , is the diagonal matrix whose th entry is . For a subgraph of , let denote the subgraph obtained from by deleting the edges of . Let and denote, respectively, the numbers of -cycles and -vertex paths in . Let denote the number of triangles containing the vertex of . Let be the union of two graphs and which have no common vertices. For any positive integer , let be the union of disjoint copies of graph . For convenience, the complete graph, path, cycle, and star on vertices are denoted by , , , and , respectively.
The permanent of matrix is defined aswhere the sum is taken over all permutations of . Valiant [1] has shown that computing the permanent is #P-complete even when restricted to (0, 1)-matrices.
Let be an matrix. The permanental polynomial of , denoted by , is defined to be the permanent of the characteristic matrix of , i.e.,where is the identity matrix of size .
Let denote the adjacency matrix of . The Laplacian matrix and signless Laplacian matrix of are defined by and , respectively. We call (resp, ) the Laplacian (resp, signless Laplacian) permanental polynomial of . The Laplacian permanental polynomial of a graph was first considered by Merris et al. [2], and the signless Laplacian permanental polynomial was first studied by Faria [3]. For more studies on (signless) Laplacian permanental polynomials, see [4–14], among others.
Two graphs and are called Laplacian copermanental if . Analogously, signless Laplacian copermanental could be defined. A graph is said to be determined by its Laplacian (resp, signless Laplacian) permanental polynomial if any graph Laplacian (resp, signless Laplacian) copermanental with is isomorphic to .
It is interesting to characterize which graph is determined by graph polynomials [15–18]. Merris et al. [2] first discussed the problem: which graph is determined by its Laplacian permanental polynomial? Answer to the problem is very hard. Up to now, only a few results are known about the problem. Merris et al. computed the Laplacian permanental polynomials of all connected graphs on 6 vertices, and they found that there exist no nonisomorphic Laplacian copermanental graphs of such graphs. Based on the result, they stated that they do not know of a pair of nonisomorphic Laplacian copermanental graphs. Recently, Liu [19] showed that complete graph and star are determined by their (signless) Laplacian permanental polynomials.
Let denote the set of graphs each of which is obtained from , by removing five or fewer edges. Cámara and Haemers [20] showed that all graphs in are determined by their characteristic polynomials of adjacency matrices of these graphs. The authors [21] proved that all graphs in are determined by their -spectra. In this paper, our interest is to discuss which graph in is detertmined by its (signless) Laplacian permanental polynomial. And, we prove the following result.
Theorem 1. All graphs in are determined by their (signless) Laplacian permanental polynomial.
The rest of this paper is organized as follows. In Section 2, we present some characterizing properties of the (signless) Laplacian permanental polynomial and give some structural properties of graphs in . In Section 3, we give the Proof of Theorem 1.
2. Preliminaries
Let denote the set of graphs each of which is obtained from by removing five or fewer edges. For , there exist exactly 45 nonisomorphic graphs each of which is obtained from by removing five or fewer edges [21, 22]. These graphs are labeled by , , and illustrated in Figure 1. For some properties of graphs in , see [21, 22], among others.
Lemma 1 (see [22]). Let be a graph with edges and let . Then,
In [22], the first author calculated the number of triangles of some , see Table 1.
Lemma 2 (see [22]). Let be a graph with edges and let . Then,In [22], the first author calculated the number of quadrangles of some , see Table 2.
Lemma 3 (see [21]). Let denote the number of triangles containing the vertex of . Using the principle of inclusion-exclusion, we can obtain the following result. Let be a graph with edges and let . Let and let be an endpoint of edges in . Then,
Lemma 4 (see [19]). Let be a graph with vertices and edges, and let be the degree sequence of . Suppose that . Then,(i)(ii)(iii)(iv)(v) +
Lemma 5 (see [19]). Let be a graph with vertices and edges, and let be the degree sequence of . Suppose that . Then,(i)(ii)(iii)(iv)(v) + By Lemmas 4 and 5, we have the following.
Corollary 1. Let be a graph with vertices and edges, and let be the degree sequence of . Suppose that and . Then,(i)(ii)For convenience, we calculate the value of some graphs in , see Table 3.
Lemma 6 (see [19]). The following can be deduced from the (signless) Laplacian permanental polynomial of a graph :(i)The number of vertices(ii)The number of edges(iii)The sum of the squares of degree of verticesWe recorded the results of the sum of squares of degrees of some graphs in advance, see Table 4.
3. The Proof of Theorem 1
We give some lemmas to prove Theorem 1 before. First, by Lemma 6 and Table 4, we obtain a result as follows.
Lemma 8. Graphs , , , , , , , , , , , , , and are determined by the (signless) Laplacian permanental polynomial, respectively.
Lemma 9. Graphs and are determined by the (signless) Laplacian permanental polynomial.
Proof. By Table 4, we know that .
By Lemma 4 (iv) and Table 1, we haveFurthermore, by Lemma 5 (v) and Tables 1–3, we haveThese imply that and are not (signless) Laplacian copermanental.
Lemma 10. The following statements hold:(i)Graphs , , and are not pairwise (signless) Laplacian copermanental(ii)Graphs and are not (signless) Laplacian copermanental(iii)Graphs and are not (signless) Laplacian copermanental
Proof. (i)By Lemma 4 (iv) and Table 1, we get that and . Furthermore, by Corollary 1 (i), Table 1, and the equations above, we obtain that , and . Furthermore, by Lemma 5 (v) and Tables 1–3, we have . These mean that graphs , , and are not pairwise (signless) Laplacian copermanental.(ii)From Table 4, we obtain that . By Lemma 4 (v) and Tables 1–3, we have . By Corollary 1 (ii), , Tables 1 and 3, and the equation above, we obtain that . This implies that and are not (signless) Laplacian copermanental.(iii)Similarly, by Lemma 4 (iv) and Table 1, we have . By Corollary 1 (i), Table 1 and the equation above, we obtain . These indicate that and are not (signless) Laplacian copermanental.
Lemma 11. The following statements hold:(i)Graphs and are not (signless) Laplacian copermanental(ii)Graphs , , , , and are not pairwise (signless) Laplacian copermanental(iii)Graphs , , , and are not pairwise (signless) Laplacian copermanental(iv)Graphs , , , and are not pairwise (signless) Laplacian copermanental(v)Graphs and are not (signless) Laplacian copermanental(vi)Graphs and are not (signless) Laplacian copermanental(vii)Graphs , , , and are not pairwise (signless) Laplacian copermanental
Proof. By Table 4, we have that(i)By Lemma 4 (v) and Tables 1–3, we have . By Corollary 1 (ii), Tables 1 and 3, and the equation above, we get that . These mean that and are not (signless) Laplacian copermanental.(ii)By Lemma 4 (iv) and (v) and Tables 1–3, we get that , , , , , , , , , and . By Corollary 1, Tables 1 and 3, and the equations above, we obtain that , , , , , , , , , and . These imply that , , , , and are not pairwise (signless) Laplacian copermanental.(iii)Similarly, by Lemma 4 (iv) and (v) and Tables 1–3, we have , , and . Furthermore, by Corollary 1, Tables 1 and 3, and the equations above, we get that , , and . These imply that graphs , , , and are not pairwise (signless) Laplacian copermanental.(iv)Similarly, by Lemma 4 (iv) and (v) and Tables 1–3, we have , , , and . By Corollary 1, Tables 1 and 3, and the equations above, we know that , , and . These mean that graphs , , , and are not pairwise (signless) Laplacian copermanental.(v)By Lemma 4 (iv) and Table 1, we get that . By Corollary 1 (i), Table 1, and the equation above, we have . These mean that graphs and are not (signless) Laplacian copermanental.(vi)By Lemma 4 (iv) and Table 1, we have that . By Corollary 1 (i), Table 1, and the equation above, we obtain that . Obviously, graphs and are not (signless) Laplacian copermanental.(vii)By Lemma 4 (iv) and Table 1, we have , , and . Furthermore, by Corollary 1, Tables 1–3, and the equations above, we have , , , and . Obviously, graphs , , , and are not pairwise (signless) Laplacian copermanental.
Proof of Theorem 1. From Lemmas 6–11, we directly obtain Theorem 1.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (no. 11761056), by Natural Science Foundation of Qinghai Province (no. 2020-ZJ-920), and the Scientific Research Innovation Team in Qinghai Nationalities University.