On minimum algebraic connectivity of graphs whose complements are bicyclic

Abstract The second smallest eigenvalue of the Laplacian matrix of a graph (network) is called its algebraic connectivity which is used to diagnose Alzheimer’s disease, distinguish the group differences, measure the robustness, construct multiplex model, synchronize the stability, analyze the diffusion processes and find the connectivity of the graphs (networks). A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. In this paper, firstly the unique graph with a minimum algebraic connectivity is characterized in the class of connected graphs whose complements are bicyclic with exactly three cycles. Then, we find the unique graph of minimum algebraic connectivity in the class of connected graphs Ωnc=Ω1,nc∪Ω2,nc, $\begin{array}{} {\it\Omega}^c_{n}={\it\Omega}^c_{1,n}\cup{\it\Omega}^c_{2,n}, \end{array}$ where Ω1,nc $\begin{array}{} {\it\Omega}^c_{1,n} \end{array}$ and Ω2,nc $\begin{array}{} {\it\Omega}^c_{2,n} \end{array}$ are classes of the connected graphs in which the complement of each graph of order n is a bicyclic graph with exactly two and three cycles, respectively.


Introduction
Let G = (V(G), E(G)) be a graph having V(G) = {v i : ≤ i ≤ n} and E(G) as the sets of vertices and edges. The graph G c is complement of G with same vertex-set and edge-set E(G c ) = {uv : u, v ∈ V(G), uv ∉ E(G)}. The number of rst neighbors of v ∈ V(G) is called its degree denoted by d (v). The adjacency matrix (A-matrix) of G is A(G) = [a i,j ]n×n such that a i,j = if v i is adjacent to v j and a i,j = otherwise. By D(G) = [a i,j ]n×n, we denote the degree matrix such that a i,i = d(v i ) and zero otherwise. The Laplacian matrix (L-matrix) of the graph G is For ≤ i ≤ n, eigenvalues µ i = µ i (G) and eigenvectors Z i = Z i (G) of L-matrix (L(G)) are the L-eigenvalues and the L-eigenvectors of G. For n-dimensional column-vectors Z i ≠ , we have L(G)Z i = µ i Z i . Since L(G) is real and symmetric therefore we have µ ≥ µ ≥ ... ≥ µ n− ≥ µn, where µn = is a minimum L-eigenvalue and µ n− (G) = a(G) is algebraic connectivity of G that remains positive if and only if G is connected. Moreover, eigenvectors corresponding to a(G) are called Fiedler vectors. For further study, we refer to [1][2][3][4][5][6][7].
The algebraic connectivity plays an important role in studies of communication and control theory to increase e ciency in air transportation system [8], measure connectivity, convergence speed & synchronization ability of the networks [9][10][11] generate or absorb the bipartition among the links [12] and construct multiplex model for the inter connected networks [13]. It is also used in brain networks to study the group di erences and complex changes in Alzheimer's disease, see [14].
A connected graph is called k-cyclic if m = n− +k, where n denotes the number of vertices, m the number of edges and k non-negative integers. In particular, if k = , , or , then G is called a tree, unicyclic, bicyclic or tricyclic graph, respectively. Let H (n, ) be a bicyclic graph with two cycles which is obtained from the star K ,n− by the addition of two edges such that each edge joins two di erent pendant vertices. Similarly, H(n, ) is a bicyclic graph with three cycles obtained from the star K ,n− by the addition of two edges in the pendant vertices such that both edges have one common end point. Let Ω ,n and Ω ,n be two classes of the bicyclic graphs with n vertices having exactly two and three cycles other than H (n, ) and H(n, ) respectively. Moreover, assume that Ω c ,n and Ω c ,n be the classes of the graphs whose complements are bicyclic with exactly two and three cycles respectively i.e Ω c n = {G c : |G c | = n and G ∈ Ω ,n } and Ω c ,n = {G c : |G c | = n and G ∈ Ω ,n }. The condition to exclude H (n, ) and H(n, ) from Ω ,n and Ω ,n respectively ensures that Ω c ,n and Ω c ,n are families of the connected graphs. Many authors studied the algebraic connectivity for di erent families of graphs such as connected graphs with certain girth, lollipop graphs and caterpillar unicyclic graphs, see [15][16][17]. Moreover, the operation of complement in graphs has important role, especially when structures of the simple graphs become more complex than their complements. Recently, Jiang et al. [18], Li et al. [19] and Javaid et al. [20,21] found the graphs with minimum algebraic connectivity among all the connected graphs whose complements are trees, unicyclic, and bicyclic with exactly two cycles. In this paper, rstly we characterize the unique graph with minimum algebraic connectivity in the class of connected graphs whose complements are bicyclic with three cycles. Then, we nd the unique graph with minimum algebraic connectivity in the complete class of connected graphs whose complements are bicyclic with two or three cycles.
The rest of the paper is managed as; In Section 2, some basic de nitions and results are given. Section 3 and Section 4 cover the main results. Section 5 includes the conclusion and some new directions of the problem.

Preliminaries
For any vector Z ∈ R n , de ne a one-one map µ : V(G) → Z such that µ(u) = Zu, where Zu is entry of Z corresponding to u ∈ V(G). Then, for Z ≠ 0, we have Moreover, if λ is a L-eigenvalue of G corresponding to Z ≠ 0 then Laplacian eigenvalue equation (LE-equation) is where Assume that Z ∈ R n is a unit vector and perpendicular to all-ones vector, then by Courant-Fisher theorem [3], we have where a(G) achieves the upper bound if Z is a Fiedler vector. If J is all-ones matrix, I is identity matrix and L(G c ) is L-matrix of the complement of G, then for any vector Z ∈ R n Suppose that C , C and C are cycles of length , and respectively. Now, some graphs are de ned which are used in the main results.
Let H and H be two bicyclic graphs with exactly three cycles which are obtained by joining any single non adjacent pair of vertices with an edge in C and C , respectively. The bicyclic graphs with exactly three cycles H is obtained from C by joining a pair of vertices with an edge such that H consists on an outer cycle of length and two inner cycles both of lengths . If we insert a vertex in an edge which is incident on two vertices of degree in H , then we obtain a bicyclic graph with three cycles H such that its all the cycles (one outer and two inner) are of lengths .  Figure 2).
Moreover, G (m , m ) is a bicyclic graph with exactly two cycles which is obtained by attaching m pendant vertices with a vertex of degree of the graph H (m + , ) (see Figure 3). Now, we state some results which are frequently used in main results.
Lemma 2.2. [18] If Z i for ≤ i ≤ n is a non-increasing sequence, then, for any ≤ i, j ≤ n,

Theorem 2.3. [21]
Let n, m and m be any positive integers such that m ≥ m ≥ , n ≥ and n = m +m + , then for any bicyclic graph with exactly two cycles G ∈ Ω ,n ,

Computational results of minimum algebraic connectivity
The computational results of the algebraic connectivity are presented in this section.
Since g( ; m , m ) = = g(a; m , m ), thus a is the second smallest root of g(λ; m , m ). Observe that Proof. Using Lemma 2.1 and (3) (as in Lemma 3.1), we nd the following polynomials of the graphs  + m a + m a + m a + m a − m a + m m a + m m a − m a − m m a − m a −  m m a − m a − a − m a − m a − m a + m a − m m a + m a + m m a + m a +  m a + m a − m a − m a − m a + m a ). Thus, Thus, Thus, Thus,

Characterization
This section consists on the main results of the paper.
Theorem 4.1. Let n, m and m be any positive integers such that m ≥ m ≥ , n ≥ and n = m + m + , then for any bicyclic graph with three cycles G ∈ Ω ,n , Proof. Let G be a bicyclic graph with three cycles C (l ), C (l ) and C (l ) with lengths l ≥ , l ≥ and l ≥ , respectively. The cycles C (l ) and C (l ) are inner cycles with at least one common edge and C (l ) is an outer cycle such that l = l + l − k, where k are common edges in C (l ) and C (l ). Let Z be a unit Fiedler vector of G c . Then, we have a sequence {Zv n } such that  (2) and Lemma 2.2, we have where Gα is a bicyclic graph with three cycles C (l ), C (l ) and C (l ) having some trees attached with the vertices of one or both the cycles C (l ) and C (l ). The lengths l , l and l may or may not di erent from l , l and l respectively. Most importantly, we note that d Gα  (1) We assume without loss of generality that both the vertices v and vn are on the cycle C (l ). Since, for l ≥ and l = the cycles C (l ) and C (l ) have two common vertices, therefore, we can assume  Figure 5). The resulting graph Gαα is a bicyclic graph with two inner cycles C (l ) and C (l ), and an outer cycle C (l ) such that l = = l , l = , some trees are attached on v in C (l ) and some trees are attached on vn which is non cycle. Thus, Gαα is a bicycle graph G (m , m ) which is obtained when we identify B by a vertex of degree with end point say v of an edge v vn having some trees on v and vn (see Figure 1 with v vn = v v ).
If (Zw m− − Zv ) < (Zw m− − Zv n ) , we delete the edge w m− w m− and add w m− vn (as (c) is obtained from (a) in Figure 5). The resulting graph Gαα is a bicyclic graph with two inner cycles C (l ) and C (l ), and an outer cycle C (l ) such that l = , l = , l = and some trees are attached on both v and vn in C (l ). Thus, Gαα is a bicycle graph which is infect B with some trees on the two adjacent vertices of degree i.e G (m , m ) (see Figure 2). If we proceed from the other side of the path, then for (Zw − Zv ) ≥ (Zw − Zv n ) , we delete the edge w w and add the edge w v otherwise, we delete w w and add w vn. Thus, the resulting graph Gαα is a bicyclic graph with three cycles such that the lengths of the inner cycles are l − and l = or l − and l = . Now, repeat the process for the vertex w and continue up to the vertex w m− . Thus, we obtain the same graphs G (m , m ) and G (m , m ).
(ii) Suppose that w i and w i+ are common vertices of the inner cycles, where ≤ i ≤ m − . If (Zw i − Zv ) ≥ (Zw i − Zv n ) , we delete w i− w i and add w i v . Now, if (Zw i+ − Zv ) ≥ (Zw i+ − Zv n ) , we delete w i+ w i+ and add w i+ v , otherwise delete w i+ w i+ and add w i+ vn. Thus, the resulting graphs are G (m , m ) or G (m , m ), respectively.
If (Zw i − Zv ) < (Zw i − Zv n ) , we delete w i− w i and add w i vn. Now, if (Zw i+ − Zv ) ≥ (Zw i+ − Zv n ) , we delete w i+ w i+ and add w i+ v , otherwise delete w i+ w i+ and add w i+ vn. Thus, the resulting graphs are G (m , m ) or G (m , m ), respectively.
(iii) Suppose that w and w are common vertices of the inner cycles, then we repeat (i) and the obtained graphs are same as there.
If in (1(i)-(iii)), l ≥ , then we can assume C (l ) = u u ...u l w i w i+ , where w i and w i+ are two common vertices of the cycles C (l ) and C (l ) for ≤ i ≤ m − and l = l − . By the use of (1(i)-(iii)), we have C (l ) with l = , some trees attached on v ∈ C (l ) and some trees attached on vn (pendant vertex) or with l = and some trees attached on v and vn (both are in C (l )). Now, for C (l ), if (Zu − Zv ) ≥ (Zu − Zv n ) , delete the edge u u and add the edge u v , otherwise delete u u and add u vn. Thus, the resulting graphs are G (m , m ) and G (m , m ) or G (m , m ) and G (m , m ), (see Figure 1 and Figure 2) respectively. Moreover, l = = l is not possible as both the vertices v and vn can not appear on only C (l ).
(2) Without loss of generality suppose that vn is on the cycle C (l ) and v is a common vertex of the inner cycles. Assume that l ≥ , l = and C (l ) = v vn w w w ...wm v , where wm is also a common vertex of the inner cycles and m = l − .
If (Zw m− − Zv ) ≥ (Zw m− − Zv n ) , we delete the edge w m− w m− and add the edge w m− v . The resulting graph Gαα is a bicyclic graph with two inner cycles C (l ) and C (l ), and an outer cycle C (l ) such that l = = l , l = , some trees are attached on v which is a common vertex of both the inner cycles and some trees are attached on vn in C (l ). Thus, Gαα is a bicycle graph G (m , m ) which is obtained when we identify B by a vertex of degree with an end point v of an edge v vn having some trees on v and vn (see Figure 1 with If (Zw m− − Zv ) < (Zw − Zv n ) , we delete the edge w m− w m− and add the edge w m− vn. The resulting graph Gαα is a bicyclic graph with two inner cycles C (l ) and C (l ), and an outer cycle C (l ) such that l = , l = , l = , some trees are attached on v which is a common vertex of both the inner cycles and some trees are attached on vn in C (l ). Thus, Gαα is a bicycle graph G (m , m ) (see Figure 1) which is infect B with some trees which are attached on two adjacent vertices of degree 2 and in C ⊆ B .
If we proceed from the other side of the path, then for (Zw − Zv ) ≥ (Zw − Zv n ) , we delete the edge w w and add the edge w v otherwise, we delete the edge w w and add the edge w vn. Thus, the resulting graph Hαα is a bicyclic graph with three cycles such that the lengths of the inner cycles are l − and l = or l − and l = . Now, repeat the process for the vertex w and continue up to the vertex w m− . Thus, we obtain the same graphs G (m , m ) and G (m , m ).
If in (2) (4) Suppose that v is on C (l ) and vn is on C (l ), where l , l ≥ . We note that d Gα (v , vn) ≥ , which is not possible.
Case b. One of v , vn is a cycle vertex. We assume that v is a cycle vertex and vn is non cycle vertex without loss of generality. In this case, for v , we have three possibilities (1) v is on C (l ), (2) v is a common vertex of both the inner cycles and (3) v is on C (l ).
(1) If v is only on C (l ), then for l ≥ and l = , we have C (l ) = v w w w ...w i w i+ ...wm v . Assume w i and w i+ are common vertices of the inner cycles, where ≤ i ≤ m − . Now, we repeat (Case a (1)) and obtain G (m , m ) or G (m , m ). If l ≥ then again by (Case a (1)), the resulting graphs are G (m , m ) and G (m , m ) or G (m , m ) and G (m , m ). Moreover, if l = = l , then the resulting graph is G (m , m ).
(2) If v is a common vertex of the inner cycles. Assume that wm is an other common vertex such that l ≥ , Then by equation (2) and Lemma (3), we have If Gαα ∉ {G i : ≤ i ≤ } and there exists a pendant vertex v, whose neighbor a is neither v nor vn, satisfying (Zv − Zv ) ≥ (Zv − Zv n ) , then delete av and add vv ; otherwise delete av and add vvn. Repeat this rearranging until the resulting graph Gααα ∈ {G i : ≤ i ≤ }.

Conclusions
In this paper, we have characterized the unique graph in the class of connected graphs whose complements are bicyclic having exactly three cycles with respect to the second least Laplacian eigenvalue (algebraic connectivity) of the Laplacian matrix. Mainly, we found the unique graph with minimum algebraic connectivity in the complete class of connected graphs whose complements are bicyclic with two or three cycles. The problem is still open to discuss the algebraic connectivity of the other families of the connected graphs whose complements are k-cyclic graphs for k ≥ (tricyclic, tetracyclic and so on.)