The Characterizing Properties of (Signless) Laplacian Permanental Polynomials of Almost Complete Graphs

<jats:p>Let <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1">
                        <mi>G</mi>
                     </math>
                  </jats:inline-formula> be a graph with <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2">
                        <mi>n</mi>
                     </math>
                  </jats:inline-formula> vertices, and let <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3">
                        <mi>L</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>G</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula> and <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4">
                        <mi>Q</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>G</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula> denote the Laplacian matrix and signless Laplacian matrix, respectively. The Laplacian (respectively, signless Laplacian) permanental polynomial of <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5">
                        <mi>G</mi>
                     </math>
                  </jats:inline-formula> is defined as the permanent of the characteristic matrix of <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6">
                        <mi>L</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>G</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula> (respectively, <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M7">
                        <mi>Q</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>G</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula>). In this paper, we show that almost complete graphs are determined by their (signless) Laplacian permanental polynomials.</jats:p>


Introduction
We use G to denote a simple graph with vertex set V(G) � v 1 , v 2 , . . . , v n and edge set E(G) � e 1 , e 2 , . . . , e m . e degree of a vertex v ∈ V(G) is denoted by d (v). e degree matrix of G, denoted by D(G), is the diagonal matrix whose (i, i)th entry is d(v i ). For a subgraph H of G, let G − E(H) denote the subgraph obtained from G by deleting the edges of H. Let c i (G) and p i (G) denote, respectively, the numbers of i-cycles and i-vertex paths in G. Let c 3 (G v ) denote the number of triangles containing the vertex v of G. Let G ∪ H be the union of two graphs G and H which have no common vertices. For any positive integer l, let lG be the union of l disjoint copies of graph G. For convenience, the complete graph, path, cycle, and star on n vertices are denoted by K n , P n , C n , and K 1,n− 1 , respectively. e permanent of n × n matrix X � (x ij ) (i, j � 1, 2, . . . , n) is defined as where the sum is taken over all permutations σ of 1, 2, . . . , n { }. Valiant [1] has shown that computing the permanent is #P-complete even when restricted to (0, 1)matrices.
Two graphs G and H are called Laplacian copermanental if π(L(G), x) � π(L(H), x). Analogously, signless Laplacian copermanental could be defined. A graph G is said to be determined by its Laplacian (resp, signless Laplacian) permanental polynomial if any graph Laplacian (resp, signless Laplacian) copermanental with G is isomorphic to G.
It is interesting to characterize which graph is determined by graph polynomials [15][16][17][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 K n and star S n are determined by their (signless) Laplacian permanental polynomials.
Let G n denote the set of graphs each of which is obtained from K n , by removing five or fewer edges. Cámara and Haemers [20] showed that all graphs in G n are determined by their characteristic polynomials of adjacency matrices of these graphs. e authors [21] proved that all graphs in G n are determined by their A α -spectra. In this paper, our interest is to discuss which graph in G n is detertmined by its (signless) Laplacian permanental polynomial. And, we prove the following result.

Theorem 1. All graphs in G n are determined by their (signless) Laplacian permanental polynomial.
e 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 G n . In Section 3, we give the Proof of eorem 1.

Preliminaries
Let G n denote the set of graphs each of which is obtained from K n by removing five or fewer edges. For n ≥ 10, there exist exactly 45 nonisomorphic graphs each of which is obtained from K n by removing five or fewer edges [21,22]. ese graphs are labeled by G ij , 1 ≤ i ≤ 5, 0 ≤ j ≤ 25, and illustrated in Figure 1. For some properties of graphs in G n , see [21,22], among others.
Lemma 1 (see [22]). Let H⊆K n be a graph with l edges and let G � K n − E(H). en, In [22], the first author calculated the number of triangles of some G ∈ G n , see Table 1.
Lemma 2 (see [22]). Let H⊆K n be a graph with l edges and let G � K n − E(H). en, In [22], the first author calculated the number of quadrangles of some G ∈ G n , see Table 2.
Lemma 3 (see [21]). Let c 3 (G v ) denote the number of triangles containing the vertex v of G. Using the principle of inclusion-exclusion, we can obtain the following result. Let H⊆K n be a graph with k edges and let G � K n − E(H). Let v ∈ V(G) and let v be an endpoint of l edges in E(H). en, Lemma 4 (see [19]). Let G be a graph with n vertices and m edges, and let Lemma 5 (see [19]). Let G be a graph with n vertices and m edges, and let (d 1 , d 2 , . . . , d n ) be the degree sequence of G.
By Lemmas 4 and 5, we have the following.   Graph Graph For convenience, we calculate the value i d i c 3 (G v i ) of some graphs in G n , see Table 3.
Lemma 6 (see [19]). e following can be deduced from the (signless) Laplacian permanental polynomial of a graph G:

(i) e number of vertices (ii) e number of edges (iii) e sum of the squares of degree of vertices
We recorded the results of the sum of squares of degrees of some graphs in advance, see Table 4.