Laplacian energy of trees with at most 10 vertices

Let Tn be the set of all trees with n ≤ 10 vertices. We show that the Laplacian energy of any tree Tn is strictly between the Laplacian energy of the path Pn and the star Sn, the authors partially proving that the conjecture hold for any tree Tn, where n ≤ 10.


Introduction and preliminaries
Let G = (V, E) be a finite, simple and undirected graph with vertices V = {1, 2, ..., n} and m = |E| edges.The degree of a vertex u ∈ V will be denoted by d n .Let G have adjacency matrix A with eigenvalues λ 1 ≥ λ 2 ≥ ... ≥ λ n , and the Laplacian matrix L = D − A, where D is the diagonal matrix of vertex degrees, with eigenvalues µ 1 ≥ µ 2 ≥ ... ≥ µ n = 0. Additional details on the theory of graph spectra may be found in [1].
The energy and Laplacian energy of G are defined as follows The energy of a graph was defined by Ivan Gutman in [2] and it has a long known chemical applications; for details see surveys [3,4].On the other hand, the Laplacian energy of a graph G, defined by Ivan Gutman and Zhou in [5].
An important problem in the area of spectral graph theory is to determine which graph G among all graphs with n vertices has the maximum energy E(G).Among the trees (a tree T is a connected undirected graph without cycles), it has been long known [7] that for positive integer n, the path P n has maximum energy and the star S n has minimum energy.For the Laplacian energy, few results are established, so that the extremal energy graphs are not known even for trees.In [8], Radenković and Gutman studied the correlation between the energy and Laplacian energy of trees.In that paper, the energy and the Laplacian energy for trees are computed.They found that the energy and the Laplacian energy of a tree are inversely proportional, and formulated the following conjecture: Conjecture 1.Let T n be a tree on n vertices.Then Furthermore, in [9], Trevisan et al., (2011) showed that the above conjecture is true for the restricted class of trees, namely those whose diameter is 3. Our goal here is to show that the Conjecture 1.1, is also true for all trees with at most 10 vertices.The plan of the paper is as follows: In the next section we will first give a notation to name trees.Next, we establish a result (Lemma 2.1) about trees with at most 10 vertices, Theorem 2.2, gives the fact that the Conjecture 1.1, is true for all trees with at most 10 vertices.We will conclude the paper with a discussion about conclusions and future work.

Names of general trees
In tables below, we need a notation to name trees.Given a tree, pick some vertex and call it the root.Now walk along the tree (depth-first), starting at the root, and when a vertex is encountered for the first time, write down its distance to the root.The sequence of integers obtained is called a level sequence for the tree.A tree is uniquely determined by any level sequence.The parent of a vertex labeled m is the last vertex encountered earlier that was labeled m − 1.For example, the tree K 1,9 gets level sequence 0111111111 if the vertex of degree 9 is chosen as root, 0122222222 otherwise.We use exponent to indicate repetition: 0111111111 can be written 01 9 and 0121212 as 0(12) 3 .Proof.If there is a tree with no edge, then it is a single vertex (it is #0 on the TABLE 1).Moreover, from the appendix (table# 2) of [1], one can find all trees T n for 2 ≤ n ≤ 10, where n is the number of vertices in T .Now we are ready to discuss the main result of the paper.
Proof.By Lemma 2.1, we have all trees with at most 10 vertices, we give all these trees in third column on the tables below in the form of level sequence.Now, from section 1, we have L = D−A the Laplacian matrix of T , and is the Laplacian energy of T .By direct calculation (one can do this exercise by computer, by use of suitable mathematical softwares, for example Matlab or Mathematica) we find the Laplacian spectrum of T (the set of eigenvalues of the matrix L, where the exponent of an eigenvalue denotes the multiplicity of the corresponding eigenvalue), and Laplacian energy of each T .Here, we note that on  4 we give all trees in such a way that the star S n appear first and the path P n appear last and all other trees appear between S n and P n for the same n.

Discussion and concluding remarks
In this paper, our attention was focused on the Laplacian energy of trees and on the partial proof of the Conjecture 1.1.We have shown that Laplacian energy of any tree T n , with n ≤ 10 vertices is strictly between the Laplacian energy of the path P n and the Laplacian energy of the star S n , a step towards the proof of the Conjecture 1.1.

Future work
let S 1,p (p ≥ 1) and S 1,q (q ≥ 1) are two stars, we introduced a special tree, denoted by T(p, q) by identifying one pendent vertex of S 1,p and one pendent vertex of S 1,q , now we see that T(p, q) has diameter 4, with n = p + q + 3 vertices; see the graph in FIGURE 2. We plan to show that the Conjecture 1.1, is hold for any tree T n such that T n ∼ = T(p, q).Moreover, from [10, p. 69-83] we find all trees T n where n ∈ {11, 12}, there are total 785 trees T n for which n ∈ {11, 12}.We also plan to show that the Conjecture 1.1, is hold for any tree T n where n ∈ {11, 12}.

TABLE 1 ,
TABLE 2, TABLE 3 and TABLE4gives a serial number, the number of vertices n, a level sequence, the Laplacian spectrum and the Laplacian energy of T .For n = 1, 2, 3 we can see fromTABLE1, that LE(P n ) = LE(T n ) = LE(S n ).For 4 ≤ n ≤ 10 from TABLE 1,

TABLE 2 ,
TABLE 3 and TABLE 4 it is easy to observe that LE(P n ) ≤ LE(T n ) ≤ LE(S n ), with equality if and only if P n = T n = S n .Laplacian energy of trees with at most 10 vertices | M. Ur Rehman, M. Ajmal and T. Kamran Laplacian energy of trees with at most 10 vertices | M. Ur Rehman, M. Ajmal and T. Kamran www.ijc.or.id www.ijc.or.id