Abstract
The algebraic connectivity of a graph is defined as the second smallest eigenvalue of the Laplacian matrix of the graph, which is a parameter to measure how well a graph is connected. In this paper, we present two unique graphs whose algebraic connectivity attain the minimum among all graphs whose complements are trees, but not stars, and among all graphs whose complements are unicyclic graphs, but not stars adding one edge, respectively.
1. Introduction
Let be a simple graph with vertex set . The complement of is denoted by , where . The adjacency matrix of is defined to be a matrix of order , where if is adjacent to , and otherwise. The degree matrix of is denoted by , where denotes the degree of a vertex in the graph . The matrix is the signless Laplacian matrix of . The Laplacian matrix of is defined to be . Since , , and are real and symmetric, their eigenvalues are real and can be arranged, respectively. We simply call the eigenvalues of as eigenvalues of , the largest eigenvalue of as spectral radius of , the eigenvalues of as the signless Laplacian eigenvalues of , and the eigenvalues of as the Laplacian eigenvalues of . The second smallest Laplacian eigenvalue of is called the algebraic connectivity of , denoted by . The eigenvectors corresponding to are called Fiedler vectors of . is a good parameter to measure how well a graph is connected and plays an important role in control theory and communications, and so forth. In particular, it is related to the synchronization ability of complex network [1, 2]. On the other hand it is related to the convergence speed in networks; one important topic is to increase as much as possible. There are several techniques to enhance this metric; see [3].
When the structures of graphs are very complex, but the structures of their complements are simple, we naturally think whether we can study the graphs by studying their complements. Recently, there are some works about this subject. A connected graph with vertices and edges is called -cyclic graph if . Specially, if , or , then is called a tree, unicyclic graph, bicyclic graph, or tricyclic graph, respectively. Let be the set of trees on vertices, and let be the set of unicyclic graphs of order . Denote by the star graph on vertices, and the graph obtained from by adding one edge . Fan et al. [4] have got the unique minimizing graph whose least eigenvalue attains the minimum among all graphs for . Li and Wang [5] have researched the unique minimizing graph which whose signless Laplacian eigenvalue attains the minimum among all graphs for . Liu and Zhang [6] have studied the unique maximizing graph whose spectral radius attains the maximum among all graphs for . Yin and Guo [7, 8] have characterized two maximizing graphs whose spectral radius attains the maximum among all the complements of bicyclic graphs of order and among all the complements of tricyclic graphs of order , respectively. Yu and Fan [9, 10] have studied two unique minimizing graphs whose least eigenvalue attains the minimum among all graphs of order whose complements are connected and whose complements are connected and have no cut edges or cut vertices, that is, 2-edge connected or 2-vertex connected graphs, respectively.
We note that the least Laplacian eigenvalue of a graph is zero, and Fiedler [11] proves that a graph is connected if and only if . In fact, is the unique graph whose complement is not connected in , and is the unique graph whose complement is not connected in . One may naturally ask what is the minimizing graph whose second smallest Laplacian eigenvalue (or the algebraic connectivity) attains the minimum among all graphs for or for ? In this paper, we study those questions and obtain two unique graphs which have the least algebraic connectivity among all graphs for and for , in Sections 3 and 4, respectively.
2. Preliminaries
We begin with some definitions. Given a graph of order , a vector is called to be defined on , if there is a 1-1 map from to the entries of , simply written for each . If is an eigenvector of , then is defined on naturally; is the entry of corresponding to the vertex . One can find that, for an arbitrary vector , and when is a Laplacian eigenvalue of corresponding to the eigenvector if and only if , where denotes the neighborhood of in the graph . Equation (2) is called the Laplacian eigenvalue-equation for the graph . In addition, by the well-known Courant-Fisher Theorem [11], for an arbitrary unit vector , when , ,with equality if and only if is a Fiedler vector of , where is null vector and is the vector such that each of its coordinates is equal to .
Let be the complement of a graph of order . It is easily seen that , where , respectively, denote the all-ones square matrix and the identity matrix both of suitable sizes. So for an arbitrary vector ,
Lemma 1 (see [11]). Let be a simple graph. Then , where .
Lemma 2. If is a nonincreasing sequence, then, for any , , .
Proof. For any , , if , by the monotone of , we have ThenSimilarly, if , we haveThenThe result follows.
3. The least Algebraic Connectivity of the Complements of Trees
Given a graph with , let be the distance between and in . Denote by a complete bipartite graph whose bipartition has vertices and vertices, respectively. Denote by the special tree, which is obtained from two disjoint stars () and () by joining the center of and by a path of length ; see Figure 1.
Lemma 3. Given a positive integer (), for any positive integers with , , one has
Proof. Let be the graph as depicted in Figure 1 with some vertices labeled. Let be a unit Fiedler vector of . By Lemma 1, we have for any vertex . Then by the Laplacian eigen-equations (2), all vertices which are the pendant vertices of have the same values given by , say ; all vertices which are the pendant vertices of have the same values given by , say . Write , . Now considering the Laplacian eigen-equations of for , and simply writing , we haveTransform (10) into a matrix equation , where and Let ; then we haveBecause and are roots of the polynomial , then is the second smallest root of the polynomial . Observe thatBy Lemma 1, we have . Then by , , and , we have that which implies that The result now follows from the above discussion.
Corollary 4. Given a positive integer (), for any positive integers with , , one has
Lemma 5. Given a positive integer (), for any tree , there exist some integers with , , , such that
Proof. Let be a unit Fiedler vector of ; then and . Thus we can get a sequence such that If , we can let the path , where when . Add the edge , and delete the edge or such that the result tree is not star. Then, by (1) and Lemma 2, we haveIf and there exists a pendant vertex , whose neighbor is neither nor , satisfyingthen delete and add ; otherwise delete and add . Repeat this rearranging until the resulting tree . By (1) and Lemma 2, we haveBy (3), (4), (19), and (21), we haveThe result follows.
By Corollary 4 and Lemma 5, we now obtain one main result of this paper.
Theorem 6. For , , one haswith equality if and only if .
4. The Least Algebraic Connectivity of the Complements of Unicyclic Graphs
Denote by the special unicyclic graph, which is obtained from by identifying two vertices with the center of and the center of , respectively; see Figure 2.
Lemma 7. Given a positive integer (), for any positive integers with , , we have
Proof. Let be the graph as depicted in Figure 2 with some vertices labeled. Let be a unit Fiedler vector of . By Lemma 1, we have for any vertex . Then, by the Laplacian eigen-equations (2), all vertices which are the pendant vertices of have the same values given by , say ; all vertices which are the pendant vertices of have the same values given by , say . Write . Now considering the Laplacian eigen-equations of for , and simply writing , we haveTransform (25) into a matrix equation , where and Let ; then we haveBecause and are roots of the polynomial , then is the second smallest root of the polynomial . Observe thatBy Lemma 1, we have . Then, by , , and , we havewhich implies thatThe result now follows from the above discussion.
Corollary 8. Given a positive integer (), for any positive integers with , , one has
Denote by the special unicyclic graph, which is obtained from by identifying one vertex with the center of and the pendent vertex of ; see Figure 3.
Lemma 9. Given a positive integer (), for any positive integers with , one has
Proof. Let be the graph as depicted in Figure 3 with some vertices labeled. Let be a unit Fiedler vector of . By Lemma 1, we have for any vertex and . Write , . Then by the Laplacian eigen-equations (2), , all vertices which are the pendant vertices of have the same values given by , namely, ; all vertices which are the pendant vertices of have the same values given by , namely, . Now considering the Laplacian eigen-equations of for , and simply writing , we haveTransform (33) into a matrix equation , where and Let ; then we haveBecause and are roots of the polynomial , then is the second smallest root of the polynomial . Observe thatBecause , we haveBy Lemma 1, we note that .
If , we have , which implies that If , we have , which implies thatIf , we haveand by Lemma 1.
LetThenand is the least root of the polynomial by . We calculate thatThus when , , , which implies that is monotonously increasing for when .
So, when , , , which implies that is monotonously decreasing for when , .
Then, if , by , we getThe result now follows from the above discussion.
Lemma 10. Given a positive integer , for any positive integers with , , one haswith equality if and only if .
Proof. From the proof of Lemma 9, we note that , are the second smallest roots of the following two polynomials, respectively:By Lemma 1 and , we have , . Thus, , are the least roots of the following two polynomials, respectively:We observe thatfor and .
Because , , and , we havewhich imply that ; namely, By Lemma 9, the result follows.
Denote by the special unicyclic graph, which is obtained from by identifying two adjacent vertices with the center of and the center of , respectively; see Figure 4.
Lemma 11. Given a positive integer (), for any positive integers with , , one has
Proof. Let be the graph as depicted in Figure 4 with some vertices labeled. Let be a unit Fiedler vector of . By Lemma 1, we have for any vertex . Write , . Then by the Laplacian eigen-equations (2), all vertices which are the pendant vertices of have the same values given by , namely, ; all vertices which are the pendant vertices of have the same values given by , namely, . Now considering the Laplacian eigen-equations of for , and simply writing , we haveTransform (52) into a matrix equation , where andLet ; then we haveBecause and are roots of the polynomial , then is the second smallest root of the polynomial . Observe thatBy Lemma 1, we have . Then by , , and , we havewhich implies that The result now follows from the above discussion.
Corollary 12. Given a positive integer (), for any positive integers with , , one has
Lemma 13. Given a positive integer (), for any unicyclic graph , there exist some integers with , such that
Proof. Let be obtained from -cycle (the cycle which has length ) by attaching some trees to it, and let be a unit Fiedler vector of . Then and . Thus we can get a sequence such thatIf , we can let the path , where when . Add the edge , and delete the edge or such that the result unicyclic graph is not . Then, by (1) and Lemma 2, we havewhere is obtained from by attaching some trees to it and .
If , we will discuss in three cases in the following.
Case 1. Both and are on the cycle .
If , we can let the cycle . In this time, if , we delete the edge and add the edge ; otherwise, we delete the edge and add the edge . Then the result unicyclic graph is obtained from ( is on the cycle) or (both and are on the cycle) by attaching some trees to it.
Case 2. One of is on the cycle .
We may let be on the cycle , and it has same discussion when is on the cycle . If , we can let the cycle . In this time, if , we delete the edge and add the edge ; otherwise, we delete the edge and add the edge . Then the result unicyclic graph is obtained from (one of is on the cycle) or (both and are on the cycle) by attaching some trees to it.
Case 3. Neither nor is on the cycle .
We can let the cycle , and neither nor is on the tree which is attached to . In this time, if , we delete the edge and add the edge ; otherwise, we delete the edge and add the edge . Then the result unicyclic graph is in Case or Case .
Then, by (1) and Lemma 2, we havewhere and is obtained from by attaching some trees to it, where at least one of is on the cycle, or is obtained from by attaching some trees to it, where both and are on the cycle.
If and there exists a pendant vertex , whose neighbor is neither nor , satisfying then delete and add ; otherwise delete and add . Repeat this rearranging until the result graph .
Then by (1) and Lemma 2, we haveAccording to (3), (4), (61), (62), and (64), we haveThe result follows.
Theorem 14. For , , one haswith equality if and only if .
Proof. From the proofs of Lemmas 7, 9, and 11, we note that , , and are the second smallest roots of the following three polynomials, respectively:By Lemma 1 and , we have . Thus, are the least roots of the following three polynomials, respectively:Claim 1. Consider .
We observe thatfor and .
Because , , and , we have which imply that ; namely,Claim 2. .
We observe thatfor
Because and , we have .
We note thatfor and .
Then is monotonously increasing for when and .
Thus by , , and , , we gain ; that is, According to Corollary 8, Lemma 10, Corollary 12, Lemma 13, Claim , and Claim , the result follows.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is jointly supported by the National Natural Science Foundation of China under Grant nos. 11071001 and 11071002, the Natural Science Foundation of Anhui Province of China under Grant no. 11040606M14, and the Natural Science Foundation of Department of Education of Anhui Province of China under Grant nos. KJ2013A196 and KJ2015ZD27.