Abstract
This research intends to construct a signless Laplacian spectrum of the complement of any -regular graph with order . Through application of the join of two arbitrary graphs, a new class of -borderenergetic graphs is determined with proof. As indicated in the research, with a regular -borderenergetic graph, sequences of regular -borderenergetic graphs can be constructed. The procedures for such a construction are determined and demonstrated. Significantly, all the possible regular -borderenergetic graphs of order are determined.
1. Introduction
All graphs considered in this paper are simple, unweighted, and undirected. Let be a graph of order , where is the vertex set of . The complement of is denoted by . The complete graph of order is denoted by . Denote the average vertex degree of by . The join of two graphs and is the graph with the vertex set and the edge set consisting of all the edges of and together with the edges joining each vertex of with every vertex of . For details on graph theory and spectral graph theory; see [1–4].
Let and be the adjacency matrix and the diagonal matrix of the vertex degrees of , respectively. Then, and are called the Laplacian matrix and the signless Laplacian matrix of , respectively. In particular, the signless Laplacian spectra of join of two regular graphs are already determined [5].
The energy of is defined as the sum of the absolute value of the eigenvalues of its adjacency matrix [6, 7]. Let be the eigenvalues of . Then,
For additional information on graph energy and its applications in chemistry, we refer to [8–10]. The eigenvalues of the Laplacian matrix of graph are denoted by . The Laplacian energy [11] of is defined as
The eigenvalues of the signless Laplacian matrix of graph are denoted by , which forms the signless Laplacian spectrum . The signless Laplacian energy of [12] is defined as .
In 2015, Gong et al. [13] proposed the concept of borderenergetic graphs, namely graphs of order satisfying . The corresponding results on borderenergetic graphs can be seen in [14–17]. For the Laplacian energy of a graph , Tura [18] proposed the concept of L-borderenergetic graphs, that is, a graph of order is -borderenergetic if . More results on -borderenergetic graphs, we can refer to [18–22].
Recently, Tao and Hou [23] extended this concept to the signless Laplacian energy of a graph. If a graph has the same signless Laplacian energy as the complete graph , i.e., , then it is called -borderenergetic. In [23, 24], several classes of -borderenergetic graphs are constructed.
Moreover, in this paper, through using the joint of two graphs, we construct a new class of -borderenergetic graphs and present a procedure to find sequences of regular -borderenergetic graphs. Especially, all regular -borderenergetic graphs of order are presented. In addition, we obtain the signless Laplacian spectrum of the complement of any -regular graph of order .
2. Construction on -Borderenergetic Graphs
At first, the signless Laplacian spectrum of the complement of any -regular graph with order is given in Lemma 1. Denote the signless Laplacian matrix of by .
Lemma 1. Let be a -regular connected graph of order . If are the eigenvalues of , then the eigenvalues of are as follows:
Proof. Note that the signless Laplacian matrix of the complement of is written aswhere is an identity matrix and is the matrix with each of whose entries is equal to 1. Since is -regular, we have that with corresponding eigenvector . Let be the eigenvectors of corresponding to the eigenvalues , respectively. Thus, we have , . Since is symmetric, all the eigenvectors are orthogonal to each other. Thus, we obtain . As can be presented asit arrives at , . Therefore,Thus, is an eigenvalue with corresponding eigenvector of , where . As is -regular, is an eigenvalue with corresponding eigenvector .
Using Lemma 1, we obtain the signless Laplacian spectrum of the join of two special graphs in the following theorem.
Theorem 1. Let be a -regular graph on vertices and be an empty graph on vertices. If are the signless Laplacian eigenvalues of , then the signless Laplacian eigenvalues of are
Proof. Note that the join of and can also be expressed withSince and are the signless Laplacian eigenvalues of and , respectively, by Lemma 1, we have that the signless Laplacian spectra of and are as follows:Thus, the set of the signless Laplacian eigenvalues of is composed of the above two sets. Using Lemma 1, we obtain the signless Laplacian eigenvalues of as follows:Using Theorem 1, from any -regular -borderenergetic graph, we can construct a new class of -borderenergetic graphs in the following theorem.
Theorem 2. Let be a -regular -borderenergetic graph with vertices. Then is -borderenergetic.
Proof. Let be the signless Laplacian eigenvalues of . Since is -borderenergetic, then we haveLet . By Theorem 1, the -spectrum of isSince , the average degree of graph isBy the definition of signless Laplacian energy of a graph with (11), we haveSince , from the above result, we conclude that is -borderenergetic.
3. Sequences of -Borderenergetic Graphs
In this section, by using Theorem 2 repeatedly, an infinite sequence of -borderenergetic graphs is constructed. Let be any -regular -borderenergetic graph with vertices. Consider an infinite sequence of graphs, i.e., such that
One can easily see that graph is of orders and -regular. And the signless Laplacian spectrum of is given in the following lemma.
Lemma 2. Let be a -regular -borderenergetic graph of order with signless Laplacian eigenvalues . Then for any , the signless Laplacian spectrum of is the following:
Proof. We prove this lemma by mathematical induction on . For , by Theorem 2, (16) holds. We now assume that the result holds for . Then we haveNow, we have . By Theorem 1, we obtainAgain (16) holds for . This completes the proof of the lemma.
Theorem 3. For any , is -borderenergetic.
Proof. Since the graph is -regular with order , by Lemma 2 and the definition of signless Laplacian energy, we haveHence, is -borderenergetic.
In fact, for a -regular graph , we can check that the three kinds of energies of are equal, i.e., .
Lemma 4. [20] If is a -regular graph, then .
Theorem 5. If is a -regular graph of order , then .
Proof. Obviously, the average degree of is . The former equality holds by Lemma 4. Moreover,This completes the proof of the theorem.
For a -regular graph of order , if is borderenergetic, then is -borderenergetic and -borderenergetic. In [13], Gong et al. found all the borderenergetic graphs with order . Bearing in mind that there are no noncomplete borderenergetic graphs with order . Furthermore, Li et al. [17] searched for the borderenergetic graphs of order 10. Thus, we can find all the regular or -borderenergetic graph of order , (Figure 1). Denote the -th -regular -borderenergetic graph of order by .
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Data Availability
The data, cited from the paper [17], used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was supported by the NSFQH (No. 2018-ZJ-925Q); NSFC (No. 11701311); and NSFGD (No. 2016A030310307).