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## Volume 7, Issue 1

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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
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  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 .
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 . The minimal Laplacian spectral radius of trees with given matching number is given by Feng et al. . The properties of spectra of graphs and their line graphs are studied by Chen . The signless Laplacian spectra of graphs is explored by Cvetković et al. . Zhou  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 . 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.  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.
 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 .

## 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.

## References

1. Nikiforov, V. Merging the A- and Q-spectral theories. Appl. Anal. Discret. Math. 2017, 11, 81–107. [Google Scholar] [CrossRef]
2. Bollobás, B. Modern Graph Theory; Springer: New York, NY, USA, 1998. [Google Scholar]
3. Xing, R.; Zhou, B. On the least eigenvalue of cacti with pendant vertices. Linear Algebra Appl. 2013, 438, 2256–2273. [Google Scholar] [CrossRef]
4. Xue, J.; Lin, H.; Liu, S.; Shu, J. On the Aα-spectral radius of a graph. Linear Algebra Appl. 2018, 550, 105–120. [Google Scholar] [CrossRef]
5. Yu, A.; Lu, M.; Tian, F. On the spectral radius of graphs. Linear Algebra Appl. 2004, 387, 41–49. [Google Scholar] [CrossRef][Green Version]
6. Ye, M.-L.; Fan, Y.-Z.; Wang, H.-F. Maximizing signless Laplacian or adjacency spectral radius of graphs subject to fixed connectivity. Linear Algebra Appl. 2010, 433, 1180–1186. [Google Scholar] [CrossRef][Green Version]
7. Feng, L.; Li, Q.; Zhang, X.-D. Minimizing the Laplacian spectral radius of trees with given matching number. Linear Multilinear Algebra 2007, 55, 199–207. [Google Scholar] [CrossRef]
8. Li, S.; Zhang, M. On the signless Laplacian index of cacti with a given number of pendant vertices. Linear Algebra Appl. 2012, 436, 4400–4411. [Google Scholar] [CrossRef][Green Version]
9. Wu, J.; Deng, H.; Jiang, Q. On the spectral radius of cacti with k-pendant vertices. Linear Multilinear Algebra 2010, 58, 391–398. [Google Scholar]
10. Lovász, L.; Peliken, J. On the eigenvalues of trees. Period. Math. Hungar. 1973, 3, 175–182. [Google Scholar] [CrossRef]
11. Chen, Y. Properties of spectra of graphs and line graphs. Appl. Math. J. Chin. Univ. Ser. B 2002, 17, 371–376. [Google Scholar]
12. Cvetković, D.; Rowlinson, P.; Simić, S.K. Signless Laplacians of finite graphs. Linear Algebra Appl. 2007, 423, 155–171. [Google Scholar] [CrossRef][Green Version]
13. Zhou, B. Signless Laplacian spectral radius and Hamiltonicity. Linear Algebra Appl. 2010, 432, 566–570. [Google Scholar] [CrossRef][Green Version]
14. Lin, H.; Zhou, B. Graphs with at most one signless Laplacian eigenvalue exceeding three. Linear Multilinear Algebra 2015, 63, 377–383. [Google Scholar] [CrossRef]
15. Nikiforov, V.; Pastén, G.; Rojo, O.; Soto, R.L. On the Aα-spectra of trees. Linear Algebra Appl. 2017, 520, 286–305. [Google Scholar] [CrossRef]
16. Berman, A.; Plemmons, R.J. Nonnegative Matrices in the Mathematical Sciences; SIAM: Philadelphia, PA, USA, 1994. [Google Scholar]
17. Collatz, L.; Sinogowitz, U. Spektrcn endlicher Graten. Abh. Math. Scm. Univ. Hamburg 1957, 21, 63–77. [Google Scholar]
18. Brualdi, R.A.; Solheid, E.S. On the spectral radius of connected graphs. Publ. Inst. Math. 1986, 39, 45–54. [Google Scholar]
19. Rowlinson, P. On the maximal index of graphs with a prescribed number of edges. Linear Algebra Appl. 1988, 110, 43–53. [Google Scholar] [CrossRef][Green Version]
20. Cvetkovic, D.; Rowlinson, P.; Simic, S. An Introduction to the Theory of Graph Spectra; Cambridge University Press: Cambridge, UK, 2009. [Google Scholar]

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