Extremal graphs for α-index

Let N(G) be the number of vertices of the graph G. Let Pl(Bi) be the tree obtained of the path Pl and the trees B1, B2, ..., Bl by identifying the root vertex of Bi with the i-th vertex of Pl. Let Vm n = {Pl(Bi) : N(Pl(Bi)) = n;N(Bi) ≥ 2; l ≥ m}. In this paper, we determine the tree that has the largest α-index among all the trees in Vm n .


Introduction
Let G be a simple undirected graph with vertex set V (G) and edge set E(G). The degree of a vertex v ∈ V (G) is d(v) or simply d v . We denote by N (G) the number of vertices of the graph G. A graph G is bipartite if there exists a partitioning of V (G) into disjoint, nonempty sets V 1 and V 2 such that the end vertices of each edge in G are in distinct sets V 1 , V 2 . In this case V 1 , V 2 are referred as a bipartition of G. A graph G is a complete bipartite graph if G is bipartite with bipartition V 1 and V 2 , where each vertex in V 1 is connected to all the vertices in V 2 . If G is a complete bipartite graph and N (V 1 ) = p and N (V 2 ) = q, the graph G is written as K p,q . The Laplacian matrix of G is the n × n matrix L(G) = D(G) − A(G), where A(G) and D(G) are the matrices adjacency and diagonal of vertex degrees of G [7], [8], and [12], respectively. It is well known that L(G) is a positive semi-definite matrix and that (0, e) is an eigenpair of L(G) where e is the all ones vector. The matrix Q(G) = A(G) + D(G) is called the signless Laplacian matrix of G (see [4], [5], and [6]). The eigenvalues of A(G), L(G) and Q(G) are called the eigenvalues, Laplacian eigenvalues and signless Laplacian eigenvalues of G, respectively. The matrices Q(G) and L(G) are positive semidefinite, (see [21]). The spectra of L(G) and Q(G) coincide if and only if G is a bipartite graph, (see [2], [4], [7], and [8]). The largest eigenvalue µ 1 of L(G) is the Laplacian index of G, the largest eigenvalue q 1 (G) of Q(G) is known as the signless Laplacian index of G and the largest eigenvalue λ 1 (G) of A(G) is the adjacency index or index of G [3].
Since A 0 (G) = A(G) and 2A 1/2 (G) = Q(G), the matrices A α (G) can underpin a unified theory of A(G) and Q(G). In this paper, the eigenvalues of the matrices A α (G) are called the α-eigenvalues of G. We write ρ α (G) for the spectral radii of the matrices A α (G) and are called the α-indices of G. The α-eigenvalue set of G is called α-spectrum of G. The spectrum of a matrix M will be denoted by Sp(M ). Let [l] denote the set {1, 2, ...l}. Given a rooted graph, define the level of a vertex to be equal to its distance to the root vertex increased by one. A generalized Bethe tree is a rooted tree in which vertices at the same level have the same degree. Throughout this paper {B i : i ∈ [l]} is a set of generalized Bethe trees. Let P l be a path of l vertices.
In this paper, we study the tree P l {B i : i ∈ [l]} obtained from P l and B 1 , B 2 , ..., B l , by identifying the root vertex of B i with the i-th vertex of P l where each B i has order greater than or equal to 2. For brevity, we write In a graph, a vertex of degree at least 2 is called an internal vertex, a vertex of degree 1 is a pendant vertex and any vertex adjacent to a pendant vertex is a quasi-pendant vertex. We recall that a caterpillar is a tree in which the removal of all pendant vertices and incident edges results in a path. We define a complete caterpillar as a caterpillar in which each internal vertex is a quasi-pendant vertex. A complete caterpillar P l (K 1,pi ) is a graph obtained from the path P l and the stars K 1,p1 , ..., K 1,p l by identifying the root of K 1,pi with the i-th vertex of P l where p i ≥ 1 for all i ∈ [l] (see Fig. 1 for an example). Let q ∈ [l]. Let A q be the complete caterpillar P l (K 1,pi ), where p q = n − 2l + 1 and p i = 1 for all i = q. Let T n,d be the class of all trees on n vertices and diameter d. Let P m be a path on m vertices and K 1,p be a star on p + 1 vertices. In [20] the authors prove that the tree in T n,d having the largest index is the caterpillar P d,n−d obtained from P d+1 on the vertices 1, 2, ..., d + 1 and the star K 1,n−d−1 identifying the root of K 1,n−d−1 with the vertex ⌈ d+1 2 ⌉ of P d+1 . In [10], for 3 ≤ d ≤ n − 4, the first ⌊ d 2 ⌋+1 indices of trees in T n,d are determined. In [9], for 3 ≤ d ≤ n−3, the first Laplacian spectral radii of trees in T n,d are characterized. In [15] the authors present some extremal results about the spectral radius ρ α (G) of A α (G) that generalize previous results about ρ 0 (G) and ρ 1/2 (G). In [24], the authors gives three edge graft transformations on A αspectral radius. As applications, we determine the unique graph with maximum A αspectral radius among all connected graphs with diameter d, and determine the unique graph with minimum A α -spectral radius among all connected graphs with given clique number. In [14] the authors gives several results about the A α -matrices of trees. In particular, it is shown that if T ∆ is a tree of maximal degree ∆, then the spectral radius of A α (T ∆ ) satisfies the tight inequality The complete caterpillars were initially studied in [18] and [19]. In particular, in [18] the authors determine the unique complete caterpillars that minimize and maximize the algebraic connectivity (second smallest Laplacian eigenvalue) among all complete caterpillars on n vertices and diameter m + 1. Below we summarize the result corresponding to the caterpillar attaining the largest algebraic connectivity. ). Among all caterpillars on n vertices and diameter m + 1, the largest algebraic connectivity is attained by the caterpillar A ⌊ m+1 2 ⌋ . Theorem 1.2 (Abreu, Lenes, Rojo [1]). Let α = 0, 1/2. Let G be a complete caterpillars on n vertices and diameter m + 1. Then, Numerical experiments suggest us that A ⌊ m+1 2 ⌋ is also the tree attaining the largest αindex in the class V m n . In this paper we prove that this conjecture is true; we come up with a bound for the whole family A α (G), which implies the result of Abreu, Lenes, and Rojo. This is organized as follows. In Section 2, we introduce trees obtained of the path P l and the trees B 1 , B 2 , ..., B l by identifying the root vertex of B i with the i-th vertex of P l and give a reduction procedure for calculating their α-spectra, thereby extending the main results of [16]. In the Section 3, we determine the graph that maximize the α-index in V m n . We finish the section maximizing the α-index among all the unicyclic connected graphs on n vertices.

The α-eigenvalues
Given a generalized Bethe tree B i with k i levels and an integer j ∈ [k i ], we write n i,ki−j+1 for the number of vertices at level j and d i,ki−j+1 for their degree. In particular, and, in particular, For i ∈ [l], it is worth pointing out that m i,1 , ..., m i,ki−1 are always positive integers, and that n i,1 ≥ n i,2 ≥ · · · ≥ n i,ki . We label the vertices of P l (B i ) as in [16]. (See figure 2). Recall that the Kronecker product C ⊗ E of two matrices C = (c i,j ) and E = (e i,j ) of sizes m × m and n × n, is an mn × mn matrix defined as which hold for any matrices of appropriate sizes. We write I l for the identity matrix of order l and j l for the column l-vector of ones. For i ∈ [l], let s i = ki−2 j=1 n i,j and D i be the matrix of order s i × l defined by Let β = 1 − α, and assume that P l (B i ) is a tree labeled as described above. It is not hard to see that the matrix A α (P l (B i )) can be represented as a symmetric block tridiagonal matrix  Let's define the polynomials P 0 (λ), P 1 (λ), ..., P l (λ) and P i,j (λ) for i ∈ [l] and j ∈ [k i ] as follows: and for i ∈ [l] and j = 2, 3, ..., k i − 1, let Moreover, let and for i = 2, 3, ..., l − 1. where .
Proof. Write A for the determinant of a square matrix A. To prove 3, we shall reduce φ(λ) = λI − A α (P l (B i )) to the determinant of an upper triangular matrix. For a start, note that and for all j ∈ [k i − 2], multiplying the j-th row of X i (λ) inserted in φ(λ) by βPi,j−1 Pi,j ⊗ j T i,mj and add it to the next row. Since we obtain, Thus, the equality (3) is proved whenever P i,j (λ) = 0 for any i ∈ [l] and j ∈ [k i − 1].
Since for any i ∈ [l] and j ∈ [k i − 1] the polynomials P i,j (λ) have finitely many roots, the equality (3) is verified for infinitely many value of λ. The proof is complete.
, let T i,j be the j ×j leading principal submatrix of the k i × k i symmetric tridiagonal matrix where β = 1 − α, c = 2 for i ∈ [l − 1] and c = 1 for i = 1 and i = l.
Since d s > 1 for all s = 2, ..., j, each matrix T j has nonzero codiagonal entries and it is known that its eigenvalues are simple. Using the well known three-term recursion formula for the characteristic polynomials of the leading principal submatrices of a symmetric tridiagonal matrix and the formulas (1) and (2), one can easily prove the following assertion: and for any i ∈ [l] and j ∈ [k i − 1].
Let A be the matrix obtained from a matrix A by deleting its last row and last column. Moreover, for i, j ∈ [r], let E i,j be the k i × k j matrix with E i,j (k i , k j ) = 1 and zeroes elsewhere. We recall the following Lemma.

Lemma 2.5 ([17]
). For i, j ∈ [r], let C i be a matrix of order k i × k i and µ i,j be arbitrary scalars. Then, From now on, for i ∈ [l − 1], by F i we denote the matrix of order k i × k i+1 whose entries are 0, except for the entry F i (k i , k i+1 ) = 1.
Proof. The characteristic polynomial of the matrix M (P l (B i )) is given by From Lemma 2.5, we have that λI − M (P l (B i )) is given by Sp(T i,j ) ∪ Sp(M (P l (B i ))); 2. the multiplicity of each eigenvalue of T i,j as an α-eigenvalue of P l (B i ) is n i,j − n i,j+1 , if i ∈ [l] and j ∈ [k i − 1], and is 1 if i ∈ [l] and j = k i ; 3. ρ α (P l (B i )) is the largest eigenvalue of M (P l (B i )); 4. ρ α (P l (B i )) > α.

The α-index of graphs
In Theorem 2.7, we characterize the α-eigenvalues of the trees P l (B i ) obtained from path P l and the generalized Bethe trees B 1 , B 2 , ..., B l obtained identifying the root vertex of B i with the i-th vertex of P l . This is the case for the caterpillars P l (K 1,pi ) in which the path is P l and each star K 1,pi is a generalized Bethe tree of 2 levels. From Theorem 2.7, we get Lemma 3.1. Let α ∈ [0, 1). Then: 1. the α-spectrum of P l (K 1,pi ) is formed by α with multiplicity l i=1 p i − l, and the eigenvalues of the 2l × 2l irreducible nonnegative matrix pi )) is the largest eigenvalue of M (P l (K 1,pi )); 3. ρ α (P l (K 1,pi )) > α.
Let t(λ, x) and s(λ, x) be the characteristic polynomials of the matrices T (x) and S(x), respectively. That is, Then, The notation A l will be used to denote the determinant of the matrix A of order l × l. The next result is an immediate consequence of the Lemma 2.5.
Proof. Let α ∈ [0, 1). Let T ∼ = P l (B i ) ∈ V m n . Let x 1 , x 2 , ..., x l be the vertices of the path P l in the tree T . Let B i be a tree with k i levels for all i ∈ [l]. Suppose T has the largest α-index in V m n . Suppose k i > 2 for some 2 ≤ i ≤ l − 1. Let u 1 , ..., u si be all the vertices in the second level of B i ; we can assume without loss of generality that u si is an internal vertex. Let w 1 , ..., w ri be all the vertices of N G (u si ) − {x i }. Let T xi ∼ = T − u si w 1 − · · · − u si w ri + x i w 1 + · · · + x i w ri , and T us i ∼ = T −x i−1 x i −x i+1 x i −u 1 x i −· · ·−u si−1 x i +x i−1 u si +x i+1 u si +u 1 u si +· · ·+u si−1 u si .
By reasoning analogously we can verify that T ∈ A n,m .