A α -spectrum of duplicate and corona operations in graphs

Let G be a graph of order n , A ( G ) its adjacency matrix and D ( G ) the diagonal matrix of degrees of G . In 2017, for every α in [0 , 1] , Nikiforov de ﬁ ned the matrix A α ( G ) = αD ( G ) + (1 − α ) A ( G ) . In this paper, we investigate the A α -spectrum of graphs obtained from the duplicate and corona operations. As an application of our results, we provide conditions for the construction of some pairs of non isomorphic A α -cospectral graphs.


Introduction
Let G = (V (G), E(G)) be a simple graph of order n and size m with vertex set V (G) and edge set E(G).The set of vertices adjacent to a vertex v ∈ V (G), denoted by N G (v), is called the neighborhood of v.The degree of a vertex v, denoted by d(v), is the cardinality of N G (v).A graph is called r-regular graph (or simply regular graph) if each of its vertices have the same degree r.
Let D(G) be the diagonal matrix of vertex degrees, and A(G) be the adjacency matrix of G.The Laplacian matrix and signless Laplacian matrix of G are defined by L(G) = D(G) − A(G) and Q(G) = A(G) + D(G), respectively.We denote the all-ones vector with n coordinates by j n and the identity matrix of order n by I n .Let M = M (G) be an n×n symmetric matrix associated with a graph G.The M -characteristic polynomial of G is defined by and its roots are called the M -eigenvalues of G or simply M -eigenvalues.
In 2017, Nikiforov [21] defined, for any real α ∈ [0, 1], the convex linear combination A α (G) of A(G) and D(G) by (G).So, it was claimed in [21,23] the matrix A α (G) can underpin a unified theory of A(G) and Q(G).For convenience, sometimes we let A α (G) = A α .The A α -characteristic polynomial of G is denoted by ϕ α (G, x) = ϕ A α (G, x), its roots are called the α-eigenvalues of G or simply α-eigenvalues, and the A α -spectrum of G is defined by σ α (G) = σ(A α ).
In [21], Nikiforov presented some elementary properties of matrix A α and proposed some open problems.In [6,22], the authors considered the semidefiniteness of the matrix A α .The extremal problems for the largest and the second largest α-eigenvalues were studied in [8,17,23,24,28,29].Many graph operations such as disjoint union, corona, edge corona, neighborhood corona, common neighborhood graphs, have been introduced and their spectral properties have been studied, see [1,3,5,9,10,12,14,15,16,18,19,26].Several variants of corona product and duplicate of two graphs have been introduced and their spectra are computed.In 2018, Adiga et al. [2] introduced duplication corona, duplication neighborhood corona and duplication edge corona of two graphs and studied their adjacency spectrum, Laplacian spectrum and signless Laplacian spectrum.In this paper, we investigate the A α -spectrum of these operations on graphs, and as an application of our results, we provide conditions for the construction of some pairs of non isomorphic A α -cospectral graphs.In Section 2, we present some definitions and preliminaries results.In the following sections we study the A α -spectrum resulting from operations duplicate, duplication corona, duplication neighborhood corona and duplication edge corona, respectively.

Preliminaries
We start this section remembering some elementary results on block matrices and Kronecker product of block matrices.Let A = [a ij ] be a m × n matrix and B = (b ij ) be a p × q matrix.The Kronecker product of A and B, denoted by A ⊗ B, is the mp × nq matrix obtained by replacing each entry a ij of A by a ij B. Lemma 2.3 ([13]).Given the matrices A, B, C and D, the following statements holds.
(i) For any scalar c, we have (cA (ii) If A and C are matrices of the same order, B and D are matrices of the same order, then (iii) If A and B are square matrices with order n and m, respectively, then Let G be a graph of order n and M be a symmetric matrix associated with G.According to [9], the M -coronal, Γ M (x), of G is defined to the sum of the entries of the matrix (xI ).If M is a square matrix of order n such that sum of entries in each row is a constant t, then Γ M (x) = n x − t .

A α -spectrum of duplicate graph
According to Sampathkumar [25], given a graph We denote by K e 3 to the graph obtained from K 3 by adding an edge.Figure 1 displays the graphs K e 3 and D u (K e 3 ).Recently, Wang et al. [27] defined the convex linear combination for any α ∈ [0, 1], and consequently Proof.The A α matrix of D u (G) can be expressed as , and the result follows. 2 According to the following corollary, when G is a regular graph the α-spectrum of D u (G) depends only of the adjacency spectrum of G.
), and we obtain the statement.
2 Remember that a graph whose spectrum consists entirely of integers is called an integral graph, see [4], and two graphs are said to be M -cospectral if they have the same M -spectrum, see [7].The following result is an immediate consequence of Theorem 3.   The following theorem provides an expression for the α-characteristic polynomial of the duplication corona operation of two graphs.

A α -spectrum of duplication corona
where λ i (M ) is an eigenvalue of matrix With suitable labeling of the vertices of G, A α (G) can be written as So, the matrix xI n 1 (n 2 +2) − A α (G), is given by From Lemma 2.1 and Lemma 2.3, we have Therefore, by Lemma 2.1, where and the result follows. 2 As a first consequence of Theorem 4.1, the following corollary, whose proof is immediate, provides conditions for the construction of A α -cospectral graphs from duplication corona operation.
The next corollary displays an expression for the A α -characteristic polynomial of G 1 • G 2 when both graphs are regular.

A α -spectrum of duplication neighborhood corona
As defined in [2], the duplication neighborhood corona of

and then joining the neighbors of v
Consider the matrices where and the result follows. 2 As a consequence of Theorem 5.1, the next result provides conditions to construct pairs of A α -cospectral graphs from the duplication neighborhood corona operation.
Corollary 5.2.Let G 1 and G 2 be regular graphs and H 1 and H 2 be any graphs.
(ii) If H 1 and H 2 are A α -cospectral graphs and The following result is an immediate consequence of the Theorem 5.1.
Corollary 5.3.Let G 1 be an r 1 -regular graph on n 1 vertices and G 2 be an r 2 -regular graph on n 2 vertices.Then where

A α -spectrum of duplication edge corona
According to [2], the duplication edge corona of , and then joining the vertices v i and v j of V 1 to every vertex in the k th copy of G 2 whenever e k = {v i , v j } ∈ E 1 .Figure 4 displays the graph K e 3 ¯K2 .
and, from Lemma 2.1, its α-characteristic polynomial is given by and the result follows. 2 As a consequence of Theorem 6.1, the next result provides conditions to construct pairs of A α -cospectral graphs from the duplication edge corona operation.Corollary 6.2.Let G 1 and G 2 be regular graphs and H 1 and H 2 be any graphs.
(i) If G 1 and G 2 are A-cospectral graphs, then G 1 ¯H1 and G 2 ¯H1 are A α -cospectral graphs.
The proof of the following result is immediate from the Theorem 6.1 and Lemma 2.1.Corollary 6.3.Let G 1 be an r 1 -regular graph on n 1 vertices and G 2 be an r 2 -regular graph on n 2 vertices.For λ i (A(G 1 )) = µ i and λ i (A(G 2 )) = λ i we have

Lemma 2 . 1 (
[13]).Let A, B, C and D be matrices.(i)If A is non-singular, then ¯A B C D ¯= |A| ¯D − CA −1 B ¯.(ii) If A and B are square matrices and AB = BA, then ¯A B C D ¯= |DA − CB| .Proposition 2.2 ([20]).Let P = " matrix.Then λ is an eigenvalue of P if and only if λ is an eigenvalue of A + B or A − B.
is the graph obtained by taking one copy of D u (G 1 ) = (V 1 ∪ W, E) and |V 1 | copies of G 2 , and then joining the vertex v i ∈ V 1 to every vertex in the i th copy of G 2 , where 1 ≤ i ≤ |V 1 |.Figure2displays the graph obtained by applying the duplication corona operation to graphs K e 3 and K 2 .

Theorem 4 . 1 .
Let G 1 and G 2 be graphs on n 1 and n 2 vertices, respectively

Corollary 4 . 4 .
Let G 1 and G 2 be regular graphs and H 1 and H 2 be any graphs.(i) If G 1 and G 2 are A-cospectral graphs, then G 1 • H 1 and G 2 • H 1 are A α -cospectral graphs.(ii) If H 1 and H 2 are A-cospectral graphs, then G 1 • H 1 and G 1 • H 2 are A α -cospectral graphs.