The A α-Spectral Radii of Graphs with Given Connectivity

The Aα-matrix is Aα(G) = αD(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 Aα(G) is said to be the Aα-spectral radius of G. In this work, we determine the graphs with largest Aα(G)-spectral radius with fixed vertex or edge connectivity. In addition, related extremal graphs are characterized and equations satisfying Aα(G)-spectral radius are proposed.


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), vu ∈ 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) = {wu ∈ E(G), u ∈ L}.For U ⊆ V(G), G[U] denote the induced subgraph of G, that is, 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 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.

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 we have the quadratic formula of X T A α (G)X can be expressed that Because A α (G) is a real symmetric matrix, and by Rayleigh principle, we have the formula 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 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] 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.
(ii) If X is a positive vector and r is a positive number such that A α (G)X < rX, 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 k n (respectively, F k n ) (k ≥ 0) be the set of such graphs with order n and vertex (resp., edge) connectivity k.Note that F 0 n = F 0 n having some disconnected graphs of order n, and 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, ∆ 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 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 0 n , then G only has two unique connected components: Proof.Denote by G a graph having the largest A α -spectral radius in F k n .
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 we would get a graph belonging to F k n (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 k n .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).
Proof.Denote by G a graph having the largest A α -spectral radius in F k n .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 k n .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 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 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.

Main Results
In this section, we will determine maximizing A α -spectral radius of of graphs with given connectivity.By Lemmas 4 and 5, we obtain the following Theorem: Theorem 1.The graph K n is the graph in F n with A α -spectral radius , and Proof.By the Lemmas 3-5, we obtain the results.
A be any matrix partitioned into blocks A ij , where A ij is an n i × n j block.Suppose that the block A ij has constant row sums b ij , and let B = (b ij ).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 Proof.Since the matrix A α = αD + (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 . 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 Note that for n = t + k + 1, that is, n − 1 = k + t.Then we have: Then we obtain that Thus, our proof is finished.Then we get: The above result is the same as [6].
Assume that w 1 joins t vertices of G 2 .Surely t ≤ min{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