Estimating the Algebraic Connectivity of Triangle-free Graphs

The second smallest eigenvalue of the Laplacian matrix of a graph is called its algebraic connectivity. In this article, we present an estimation for the algebraic connectivity of a triangle-free graph. Then, we give some examples to illustrate our results and make a comparison with some known lower bounds.


Introduction
Let G=(V, E) be a simple connected graph with v The Laplacian matrix of G is defined as , where A is the adjacency matrix of G and D is the diagonal matrix of vertex degrees.Clearly, L is symmetric and positive semidefinite and consequently its eigenvalues, which are called the Laplacian eigenvalues of G, are real nonnegative numbers and since each row sum of L is equal to 0, then 0 is the smallest eigenvalue of L. Denote the eigenvalues of L by The spectrum of the Laplacian matrix is important in graph theory because it has a close relation to numerous graph invariants, such as diameter, chromatic number, maximum cut, spanning trees, connectivity, etc. ( for more details, see [2,3] and the references therein).Thus, good lower and upper bounds and approximations for the eigenvalues of L are needed in many applications.Among them is


This fact led Fiedler [2,3] to define the algebraic connectivity of G to be equal to

). (G a
There where attempts to find a lower bound for ) (G a in terms of simple properties of graphs such as diameter, which we denote here by  , and the number of vertices and edges of the graph ( see [1] for a comprehensive review).Among the known lower bounds for the agebraic connectivity, (2) Lu, Zhang and Tian bound [7] In this paper, we present an estimation for the algebraic connectivity of trianglefree graphs based on degrees of vertices.Then we apply our results to some examples of triangle-free graphs and make a comparison with the above mentioned bounds and refer to some known results to conlude that our estimations are good in some sense.

Main Results
We will employ the following result, known as the Cauchy interlacing property.For a proof, see [5, p. 189].
In the sequel, we assume that G is a triangle-free graph.For a vertex  be the vertices adjacent to v, where . Since G is triangle-free, then it follows that . Then by applying Lemma 1 to L and its principal submatrix v L , we have and applying Lemma 1 to v L and its principal submatrix v D yields Also, by the ordering of the eigenvalues of L , we have that So, from (3) and ( 5), we note that for each . It can be shown [6, p. 411] that the characteristic polynomial of We first consider the case in (4) where Theorem 1.Let v g be the function defined for Then, the minimal zero of v g , say v  , is less than or equal to the minimal zero of Proof .Form (6), we have that for all and since implies which (8).
From (7), the equation It is noted that the results above are proved under the assumption that In the the following lemma, we show that the inequality (8) also holds when then in view of (3), ( 5) and ( 8) one can consider  as an estimation for the algebraic connectivity ) (G a .Corollary 1.For a star graph G we have  is a lower bound for the algebraic connectivity and For any star graph G, we have It can be easily shown that and that

Examples
We now give some examples to illustrate our results.Consider the two triangle-free graphs shown in the figure below.

Fig. 1
The actual value of the algebraic connectivity , ) (G a , the lower bounds in (1) and ( 2) and the estimation  in (9) are shown in the table below.Table 1 4. Remarks (1) Table1 shows that in some cases,  is a better estimation for the algebraic connectivity than the lower bounds in ( 1) and ( 2).
(2) For star graphs , it can be easily seen that as , both the lower bounds in ( 1) and ( 2) get further from 1 while, as stated in Corollary 1, our estimation ( , respectively.This adds an example where our estimation in (9) is better than some known lower bounds for the algebraic connectivity.
(4) It is known [4] that random k-regular graphs have the algebraic connectivity bounded away from zero.This shows that the lower bounds in (1) and ( 2) are poor for regular graphs.In fact, the authors in [4] proved that for a random k-regular graph G, with probability 1  as n grows.Hence, the result of Corollary 3 is good in some sense.
(5) Finally, we would like to note that  is just an estimation for the algebraic connectivity which may be far from the true value for some graphs.For example, consider the k-cube which is bipartite (hence triangle-free), k-regular, and has algebraic connectivity equal to 2 regardless of the value of k , so corollary 3 does not apply for k-cubes with large k.
let v d denote the degree of .


Usually, the algebraic connectivity of a graph G is denoted by

Corollary 3 .
, as in(10), is larger than the lower bounds in (1) and (2).Let G be a k-regular triangle-free graph, then an estimation for the algebraic connectivity is It can be easily shown that for a k-regular triangle-free graph, For a star graph, it is easy to see that the lower bounds in (1) and (2) are