Construction for the Sequences of Q-Borderenergetic Graphs

School of Mathematics and Statistics, Qinghai Normal University, Xining 810001, China Academy of Plateau Science and Sustainability, Xining 810016, China Key Laboratory of Tibetan Information Processing, Ministry of Education, Xining 810008, China Tibetan Intelligent Information Processing and Machine Translation Key Laboratory, Qinghai, Xining 810008, China Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea


Introduction
All graphs considered in this paper are simple, unweighted, and undirected. Let G be a graph of order n � |V(G)|, where V(G) is the vertex set of G. e complement of G is denoted by G. e complete graph of order n is denoted by K n . Denote the average vertex degree of G by d. e join of two graphs H 1 and H 2 is the graph H 1 ∇H 2 with the vertex set V(H 1 )∪V(H 2 ) and the edge set consisting of all the edges of H 1 and H 2 together with the edges joining each vertex of H 1 with every vertex of H 2 . For details on graph theory and spectral graph theory; see [1][2][3][4].
Let A(G) and D(G) be the adjacency matrix and the diagonal matrix of the vertex degrees of G, respectively. en, L(G) � D(G) − A(G) and Q(G) � D(G) + A(G) are called the Laplacian matrix and the signless Laplacian matrix of G, respectively. In particular, the signless Laplacian spectra of join of two regular graphs are already determined [5].
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 K n , i.e., QE(G) � QE(K n ) � 2(n − 1), then it is called Q-borderenergetic. In [23,24], several classes of Q-borderenergetic graphs are constructed.
Moreover, in this paper, through using the joint of two graphs, we construct a new class of Q-borderenergetic graphs and present a procedure to find sequences of regular Q-borderenergetic graphs. Especially, all regular Q-borderenergetic graphs of order 7 < n ≤ 10 are presented. In addition, we obtain the signless Laplacian spectrum of the complement of any k-regular graph G of order n.

Construction on Q-Borderenergetic Graphs
At first, the signless Laplacian spectrum of the complement of any k-regular graph G with order n is given in Lemma 1. Denote the signless Laplacian matrix of G by Q.

Lemma 1.
Let G be a k-regular connected graph of order n. If μ 1 ≥ μ 2 ≥ · · · ≥ μ n are the eigenvalues of Q(G), then the eigenvalues of Q(G) are as follows: Proof. Note that the signless Laplacian matrix of the complement G of G is written as where I is an identity matrix and J is the matrix with each of whose entries is equal to 1. Since G is k-regular, we have that it arrives at Jx i � 0, i � 2, . . . , n. erefore, us, n − 2 − μ i is an eigenvalue with corresponding eigenvector x i of Q(G), where i � 2, . . . , n. As G is (n − 1 − k)-regular, 2(n − 1 − k) is an eigenvalue with corresponding eigenvector e � (1, . . . , 1) T .
Using Lemma 1, we obtain the signless Laplacian spectrum of the join of two special graphs in the following theorem. □ Theorem 1. Let G 1 be a k-regular graph on n vertices and G 2 be an empty graph on n − k vertices. If 2k � μ 1 ≥ μ 2 ≥ · · · ≥ μ n are the signless Laplacian eigenvalues of G 1 , then the signless Laplacian eigenvalues of G 1 ∇G 2 are Proof. Note that the join of G 1 and G 2 can also be expressed with Since 2k � μ 1 ≥ μ 2 ≥ · · · ≥ μ n and 0 (n− k) are the signless Laplacian eigenvalues of G 1 and G 2 , respectively, by Lemma 1, we have that the signless Laplacian spectra of G 1 and G 2 are as follows: us, the set of the signless Laplacian eigenvalues of G 1 ∪ G 2 is composed of the above two sets. Using Lemma 1, we obtain the signless Laplacian eigenvalues of G 1 ∇G 2 as follows: n − k + μ 2 , n − k + μ 3 , . . . , n − k + μ n , n (n− k− 1) , k, 2n. (10) Using eorem 1, from any k-regular Q-borderenergetic graph, we can construct a new class of Q-borderenergetic graphs in the following theorem.

Theorem 2. Let G be a k-regular Q-borderenergetic graph with n vertices. en G∇K n− k is Q-borderenergetic.
Proof. Let 2k � μ 1 ≥ μ 2 ≥ · · · ≥ μ n be the signless Laplacian eigenvalues of G. Since G is Q-borderenergetic, then we have Let p � n − k. By eorem 1, the Q-spectrum of G∇K p is Spec Q G∇K p � p + μ 2 , p + μ 3 , ... ,p + μ n , n, ... , n √√ √√ Since p � n − k, the average degree d of graph G∇K p is By the definition of signless Laplacian energy of a graph with (11), we have

(14)
Since |V(G∇K p )| � 2n − k, from the above result, we conclude that G∇K p is Q-borderenergetic.

(15)
One can easily see that graph G (s) (s � 1, 2, . . .) is of orders n + s(n − k) and n + (s − 1)(n − k)-regular. And the signless Laplacian spectrum of G (s) is given in the following lemma.

Lemma 2.
Let G (0) be a k-regular Q-borderenergetic graph of order n with signless Laplacian eigenvalues 2k � μ 1 ≥ μ 2 ≥ · · · ≥ μ n− 1 ≥ μ n . en for any G (s) ∈ H(s ≥ 1), the signless Laplacian spectrum of G (s) is the following: Proof. We prove this lemma by mathematical induction on s. For s � 1, by eorem 2, (16) holds. We now assume that the result holds for s � t. en we have Now, we have G t+1 � G t ∇K n− k . By eorem 1, we obtain  Proof. Since the graph G (s) is n + (s − 1)(n − k)-regular with order n + s(n − k), by Lemma 2 and the definition of signless Laplacian energy, we have Hence, G (s) is Q-borderenergetic. In fact, for a k-regular graph G, we can check that the three kinds of energies of G are equal, i.e.,

Theorem 5.
If G is a k-regular graph of order n, then Proof. Obviously, the average degree of G is k. e former equality holds by Lemma 4. Moreover, is completes the proof of the theorem. For a k-regular graph of order n, if G is borderenergetic, then G is Q-borderenergetic and L-borderenergetic. In [13], Gong et al. found all the borderenergetic graphs with order 7 ≤ n ≤ 9. Bearing in mind that there are no noncomplete borderenergetic graphs with order n < 7. Furthermore, Li et al. [17] searched for the borderenergetic graphs of order 10. us, we can find all the regular Q or L-borderenergetic graph of order n, 7 ≤ n ≤ 10 ( Figure 1). Denote the i-th k-regular Q-borderenergetic graph of order n by G i n,k . □ Data Availability e data, cited from the paper [17], used to support the findings of this study are included within the article.  Figure 1: e regular Q-borderenergetic graphs (a) G 1 8,5 , (b) G 2 9,4 , (c) G 3 9,4 , (d) G 4 10,7 , (e) G 5 10,6 , and (f ) G 6 10,6 .