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Mathematics 2019, 7(1), 44; https://doi.org/10.3390/math7010044

Article
The Aα-Spectral Radii of Graphs with Given Connectivity
1,† and 2,*,†
1
School of Mathematics and Statistics and Hubei key Laboratory Mathematics Sciences, Central China Normal University, Wuhan 430079, China
2
Department of Mathematics, Savannah State University, Savannah, GA 31419, USA
*
Correspondence: [email protected]; Tel.: +1-352-665-3381
These authors contributed equally to this work.
Received: 22 November 2018 / Accepted: 24 December 2018 / Published: 4 January 2019

Abstract

:
The A α -matrix is A α ( G ) = α D ( G ) + ( 1 α ) A ( G ) with α [ 0 , 1 ] , given by Nikiforov in 2017, where A ( G ) is adjacent matrix, and D ( G ) is its diagonal matrix of the degrees of a graph G. The maximal eigenvalue of A α ( G ) is said to be the A α -spectral radius of G. In this work, we determine the graphs with largest A α ( G ) -spectral radius with fixed vertex or edge connectivity. In addition, related extremal graphs are characterized and equations satisfying A α ( G ) -spectral radius are proposed.
Keywords:
adjacent matrix; signless Laplacian; spectral radius; connectivity

1. Introduction

We consider simple finite connected graph G with the vertex set V ( G ) and the edge set E ( G ) . The number of vertices | V ( G ) | = n is the order of a graph, and the number of edges | E ( G ) | is the size of a graph. Denote the neighborhood of v V ( G ) by N ( v ) = { u V ( G ) , v u E ( G ) } , and the degree of v by d G ( v ) = | N ( v ) | (or briefly d v ). For L V ( G ) and R E ( G ) , let w ( G L ) or w ( G R ) be the number of components of G L or G R . L ( or R ) be a vertex(edge) cut set if w ( G L (or R ) ) 2 and E ( w , L ) = { w u E ( G ) , u L } . For U V ( G ) , G [ U ] denote the induced subgraph of G, that is, V ( G [ U ] ) = U and E ( G [ U ] ) = { u v | u v E ( G ) , u , v U } .
If A ( G ) is adjacency matrix of a graph G, and D ( G ) is its diagonal matrix of the degrees of G, then the signless Laplacian matrix of G is D ( G ) + A ( G ) . With the successful studies of these matrices, Nikiforov [1] proposed the A α -matrix
A α ( G ) = α D ( G ) + ( 1 α ) A ( G )
with α [ 0 , 1 ] . Obviously, A 0 ( G ) is the adjacent matrix and A 1 2 is the half of signless Laplacian matrix of G, respectively. For undefined terminologies and notations, we refer to [2].
The research of (adjacency, signless Laplacian) spectral radius is an intriguing topic during past decades [3,4,5,6,7,8,9]. For instances, Lovász and J. Pelikán studied the spectral radius of trees [10]. The minimal Laplacian spectral radius of trees with given matching number is given by Feng et al. [7]. The properties of spectra of graphs and their line graphs are studied by Chen [11]. The signless Laplacian spectra of graphs is explored by Cvetković et al. [12]. Zhou [13] found bounds of signless Laplacian spectral radius and its hamiltonicity. Graphs having none or one signless Laplacian eigenvalue larger than three are obtained by Lin and Zhou [14]. At the same time, the maximal adjacency or signless Laplacian spectral radius have attracted many interests among the mathematical literature including algebra and graph theory. Ye et al. [6] gave the maximal adjacency or signless Laplacian spectral radius of graphs subject to fixed connectivity.
Inspired by these outcomes, we determine the graphs with largest A α ( G ) -spectral radius with given vertex or edge connectivity. In addition, the corresponding extremal graphs are provided and the equations satisfying the A α ( G ) -spectral radius are obtained.

2. Preliminary

In this section, we provide some important concepts and lemmas that will be used in the main proofs.
Denote by G a graph such that V ( G ) = { v 1 , v 2 , , v n } is its vertex set and E ( G ) is its edge set. The A α -matrix of G has the ( i , j ) -entry of A α ( G ) is 1 α if v i v j E ( G ) ; α d ( v i ) if i = j , and otherwise 0. For α [ 0 , 1 ] , let λ 1 ( A α ( G ) ) λ 2 ( A α ( G ) ) λ n ( A α ( G ) ) be the eigenvalues of A α ( G ) . The A α -spectral radius of G is considered as the maximal eigenvalue ρ : = λ 1 ( A α ( G ) ) . Let X = ( x v 1 , x v 2 , , x v n ) T be a real vector of ρ .
By A α ( G ) = α D ( G ) + ( 1 α ) A ( G ) , we have the quadratic formula of X T A α ( G ) X can be expressed that
X T A α ( G ) X = α v i V ( G ) x v i 2 d v i + 2 ( 1 α ) v i v j E ( G ) x v i x v j .
Because A α ( G ) is a real symmetric matrix, and by Rayleigh principle, we have the formula
ρ ( G ) = m a x X 0 X T A α ( G ) X X T X .
As we know that once X is an eigenvector of ρ ( G ) for a connected graph G, X should be unique and positive. The corresponding eigenequations for A α ( G ) is rewritten as
ρ ( G ) x v i = α d v i x v i + ( 1 α ) v i v j E ( G ) x v j .
As A 1 ( G ) = D ( G ) , we study the A α -matrix for α [ 0 , 1 ) below. Based on the definition of A α -spectral radius, we have
Lemma 1.
[4,15] Let A α ( G ) be the A α -matrix of a connected graph G ( α [ 0 , 1 ) ) , v , w V ( G ) , u T V ( G ) such that T N ( v ) \ ( N ( w ) { w } ) . Let G * be a graph with vertex set V ( G ) and edge set E ( G ) \ { u v , u T } { u w , u T } , and X a unit eigenvector to ρ ( A α ( G ) ) . If x w x v and | T | 0 , then ρ ( G * ) > ρ ( G ) .
If G is a connected graph, then A α ( G ) is a nonnegative irreducible symmetric matrix. By the results of [1,16,17] and adding extra edges to a connected graph, then A α -spectral radius will increase and the following lemma is straightforward.
Lemma 2.
( i ) If G * is any proper subgraph of connected graph G, and ρ is the A α -spectral radius, then ρ ( G * ) < ρ ( G ) .
( i i ) If X is a positive vector and r is a positive number such that A α ( G ) X < r X , then ρ ( G ) < r .
Recall that the vertex connectivity (respectively, edge connectivity) of a graph G is the smallest number of vertices (respectively, edges) such that if we remove them, the graph will be disconnected or be a single vertex. For convenience, let F n be the set of all graphs of order n, and F n k (respectively, F ¯ n k ) ( k 0 ) be the set of such graphs with order n and vertex (resp., edge) connectivity k. Note that F n 0 = F ¯ n 0 having some disconnected graphs of order n, and F n n 1 = F ¯ n n 1 consisting of the unique graph K n . Obviously, F n = k F n k = k F ¯ n k .
Recall the graph K ( p , q ) ( p q 0 ) obtained from K p by attaching a vertex together with edges connecting this vertex to q vertices of K p . K ( p , q ) is was found by Brualdi and Solehid in terms of stepwise adjacency matrix, but it is Peter Rowlinson who gives the purely combinatorial definition of such graph. For the property of K ( p , q ) , we refer to [18,19,20]. Clearly, K ( p , 0 ) is K p with an additional isolated vertex. It’s not hard to see that K ( p , q ) is of vertex (resp., edge) connectivity q. Let δ , Δ be the smallest and largest degrees of vertices in the graph G, respectively.
Lemma 3.
The graph K n is the graph in F n having the largest A α -spectral radius, and K n 1 K 1 = K ( n 1 , 0 ) is the graph in F n 0 or F ¯ n 0 having the smallest A α -spectral radius.
Proof. 
By Lemma 2, the first statement is clear. For the second one, let G be a graph which attains the maximum A α -spectral radius in F n 0 , then G only has two unique connected components: K n 1 , K 1 ; if not, any component of G will be a proper subgraph of K n 1 . Then ρ ( G ) < ρ ( K n 1 ) = ρ ( K n 1 K 1 ) , a contradiction. Then this lemma is proved. □
Lemma 4.
For k [ 1 , n 2 ] , K ( n 1 , k ) is the graph having the largest A α -spectral radius in F n k .
Proof. 
Denote by G a graph having the largest A α -spectral radius in F n k . x is a unit (positive) Perron vector of A α . Let U be the vertex cut of G having k vertices, and these components of G U be G 1 , G 2 , , G s , for s 2 . We declare that s = 2 ; if not, adding all possible edges within the graph G 1 G 2 G s 1 , we would get a graph belonging to F n k (because U is the smallest vertex cut set) and with a larger A α -spectral radius. Similarly, induced subgraph G [ U ] , the subgraphs G 1 and G 2 are complete subgranph, and every vertex of U connects these vertices of G 1 and G 2 . Next we prove that one of G 1 , G 2 will be a singleton, which has a unique vertex. If not, suppose that G 1 , G 2 have orders greater than one. Without loss of generality, denote by u a vertex of G 1 having a smallest value for x among vertices in G 1 G 2 . Deleting these edges of G 1 incident to u, and connecting all possible edges between G 1 u and G 2 , we get a graph G ˜ = K ( n 1 , k ) still in F n k . By Lemma 1, ρ ( G ˜ ) > ρ ( G ) , which yields a contradiction. So one of G 1 , G 2 is a singleton, and G is the desired graph K ( n 1 , k ) . □
Lemma 5.
For k [ 1 , n 2 ] , K ( n 1 , k ) is the graph having maximum A α -spectral radius in F ¯ n k .
Proof. 
Denote by G a graph having the largest A α -spectral radius in F n k . x is a unit (positive) Perron vector of A α . We know that each vertex of G has degree greater than or equal to k. Otherwise G F ¯ n k . If there is a vertex u in G with degree k, then the edges adjacent to u are an edge cut such that G u is complete. The statement follows in this case. Then we will suppose that all vertices in G have degrees greater than k. Let E c be an edge cut set of G having k edges. So G E c consists of only two components G 1 , G 2 , respectively, of order n 1 , n 2 . Obviously G 1 , G 2 are both complete. In addition, neither of G 1 , G 2 is a singleton. Otherwise G would contain a vertex of degree k, which contradicted to the above assumption. So G 1 , G 2 contain more than 1 vertex, i.e., n 1 2 and n 2 2 .
Without loss of generality, suppose that G 1 contains a vertex w 1 having a minimal value given by x within all vertices of G 1 G 2 , and consists of vertices w 1 , w 2 , , w n 1 such that x ( w 1 ) x ( w 2 ) x ( w n 1 ) . Assume that w 1 joins t vertices of G 2 . Surely t m i n { k , n 2 } .
If t = k , there exist no edges joining G 1 w 1 and G 2 , and n 2 k + 2 otherwise G 2 contains a vertex of degree k. Denote by G a new graph with vertex set V ( G ) and edge set E ( G ) \ E ( w 1 , N ) E ( N , v ) , where N = N ( w 1 ) V ( G 1 ) , and v V ( G 2 ) N ( w 1 ) V ( G 2 ) , by Lemma 1, we have ρ ( G ) > ρ ( G ) . Let G be another new graph with vertex set V ( G ) and adding all possible edges between G 1 w 1 and G 2 . Note that G = K ( n 1 , k ) , and G is a proper subgraph of G . By Lemma 2, we have ρ ( G ) > ρ ( G ) . Thus, ρ ( G ) > ρ ( G ) , a contradiction.
If t < k . Partition the set V ( G 1 ) w 1 as: V 11 = { w i : i = 2 , 3 , , n 1 ( k t ) } , V 12 = { w j : j = n 1 ( k t ) + 1 , , n 1 } . Thus, | V 11 | = n 1 ( k t ) 1 ; | V 12 | = k t .
Let N = N ( w 1 ) V 11 , then N since d ( w 1 ) > k . Note there is vertex v V ( G 2 ) N ( w 1 ) V ( G 2 ) since n 2 k + 2 . Let G be a new graph having vertex set V ( G ) and edge set E ( G ) \ E ( w 1 , N ) E ( N , v ) , where N = N ( w 1 ) V 11 , and v V ( G 2 ) N ( w 1 ) V ( G 2 ) , by Lemma 1, we have ρ ( G ) > ρ ( G ) . Let G be another new graph having vertex set V ( G ) and adding all possible edges between G 1 w 1 and G 2 , adding all edges between w 1 and V 12 . Note that G = K ( n 1 , k ) , and G is a proper subgraph of G . Lemma 2 implies that ρ ( G ) > ρ ( G ) . Thus, ρ ( G ) > ρ ( G ) , a contradiction. The result follows. □

3. Main Results

In this section, we will determine maximizing A α -spectral radius of of graphs with given connectivity. By Lemma 4 and Lemma 5, we obtain the following Theorem:
Theorem 1.
The graph K n is the graph in F n with A α -spectral radius, and K n 1 K 1 = K ( n 1 , 0 ) is the unique one in F n 0 or F ¯ n 0 with A α -spectral radius. For k [ 1 , n 2 ] , K ( n 1 , k ) is the graph with maximum A α -spectral radius in F n k or F ¯ n k .
Proof. 
By the Lemmas 3–5, we obtain the results. □
Lemma 6.
[20] Given a partition { 1 , 2 , , n } = Δ 1 Δ 2 Δ m with | Δ i | = n i > 0 , A be any matrix partitioned into blocks A i j , where A i j is an n i × n j block. Suppose that the block A i j has constant row sums b i j , and let B = ( b i j ) . Then the spectrum of B is contained in the spectrum of A (taking into account the multiplicities of the eigenvalues).
Since K ( n 1 , k ) contains K n 1 , we can partition K ( n 1 , k ) into three different subsets: { u } , T , S , in which u is the vertex connecting a complete subgraph K n 1 with k edges, a subset S is in K n 1 connecting u, and T = V ( K n 1 \ S ) . Let x be a Perron vector of K ( n 1 , k ) . S = { u 1 , u 2 , , u k } and T = { v 1 , v 2 , , v t } . Note that k + t + 1 = n .
Theorem 2.
Label the vertices of K ( n 1 , k ) as u , u 1 , u 2 , , u k , v 1 , v 2 , v t with k , t 0 . The maximum eigenvalues of A α ( K ( n 1 , k ) ) satisfy the equation: f ( ρ ) = ( ρ k α ) ( ρ k α n + k + 2 ) ( ρ n α + 1 ) k ( 1 α ) ( ρ k α α + 1 ) ( ρ n α + α + 1 ) + k ( 1 α ) 3 ( n k 1 ) = 0 .
Proof. 
Since the matrix A α = α D + ( 1 α ) A , where D has on the diagonal the vector ( k , n 1 , n 2 ) and A consists of the following three row-vectors, in the order: ( 0 , k , 0 ) ; ( 1 , k 1 , n k 1 ) ; ( 0 , k , n k 2 ) . Thus, by the Lemma 6, x is a constant value β 2 on the vertex set S, and constant value β 3 on the vertex set T. Defining x ( u ) = : β 1 , ρ ( K ( n 1 , k ) ) = : ρ , also by (1), we get
( ρ α k ) β 1 = k ( 1 α ) β 2
( ρ α ( n 1 ) ) β 2 = ( 1 α ) ( β 1 + ( k 1 ) β 2 + t β 3 ) , and
( ρ α ( n 2 ) ) β 3 = ( 1 α ) ( k β 2 + ( t 1 ) β 3 ) .
Then we get
( ρ α ( n 1 ) ) = k ( 1 α ) 2 ρ k α + k t ( 1 α ) 2 ρ k α t + 1 + ( k 1 ) ( 1 α ) .
Note that for n = t + k + 1 , that is, n 1 = k + t . Then we have:
( ρ k α ) = k ( 1 α ) 2 ρ k α + k t ( 1 α ) 2 ρ k α t + 1 + ( k 1 ) ( 1 α ) + t α .
Then we obtain that
( ρ k α ) ( ρ k α n + k + 2 ) ( ρ n α + 1 ) k ( 1 α ) ( ρ k α α + 1 ) ( ρ n α + α + 1 ) + k ( 1 α ) 3 ( n k 1 ) = 0 .
Thus, our proof is finished. □
Corollary 1.
Let G be a graph of order n having vertex/edge connectivity k, where 1 k n 2 , the maximum adjacency spectral radius is the largest root of the f ( λ ) = λ 3 ( n 3 ) λ 2 ( n + k 2 ) λ + k ( n k 2 ) = 0 .
Proof. 
By Theorem 2, let α = 0 , then f ( λ ) = λ 3 ( n 3 ) λ 2 ( n + k 2 ) λ + k ( n k 2 ) = 0 . It is obvious since A 0 = A ( G ) . □
By letting the special values for α , we have the following corollary.
Corollary 2.
Let G be a graph of order n having vertex/edge connectivity k, where 1 k n 2 , the signless Laplacian spectral radius λ 1 = 2 n + k 4 + ( 2 n k 4 ) 2 + 8 k 2 .
Proof. 
By Theorem 2, let α = 1 2 , then f ( λ ) = λ 3 1 2 ( 3 n + k 6 ) λ 2 + ( 1 4 ( n 4 ) ( 2 n + 3 k ) + k + 2 ) λ 1 4 k ( n 2 5 n + 6 ) = 0 . It is obvious since 2 A 1 2 = D + Q . Thus,
8 f ( λ ) = 8 [ λ 3 1 2 ( 3 n + k 6 ) λ 2 + ( 1 4 ( n 4 ) ( 2 n + 3 k ) + k + 2 ) λ 1 4 k ( n 2 5 n + 6 ) ] = ( 2 λ ) 3 ( 3 n + k 6 ) ( 2 λ ) 2 + ( ( n 4 ) ( 2 n + 3 k ) + 4 k + 8 ) ( 2 λ ) 2 k ( n 2 5 n + 6 ) = ( λ 1 ) 3 ( 3 n + k 6 ) ( λ 1 ) 2 + ( ( n 4 ) ( 2 n + 3 k ) + 4 k + 8 ) ( λ 1 ) 2 k ( n 2 5 n + 6 ) .
Let λ 1 = 2 λ and
F ( λ 1 ) = ( λ 1 ) 3 ( 3 n + k 6 ) ( λ 1 ) 2 + ( ( n 4 ) ( 2 n + 3 k ) + 4 k + 8 ) ( λ 1 ) 2 k ( n 2 5 n + 6 ) = 0 .
Then we get:
λ 1 = 2 n + k 4 + ( 2 n k 4 ) 2 + 8 k 2 .
 □
The above result is the same as [6].

Author Contributions

All authors have contributed equally to this work. Investigation and Methodology: C.W.; Methodology and Correction: S.W.

Funding

This work was partially supported by the National Natural Science Foundation of China under Grants 11771172 and 11571134.

Conflicts of Interest

The authors declare no conflict of interest.

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