New Bounds for the α -Indices of Graphs

: Let G be a graph, for any real 0 ≤ α ≤ 1, Nikiforov deﬁnes the matrix A α ( G ) as A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) , where A ( G ) and D ( G ) are the adjacency matrix and diagonal matrix of degrees of the vertices of G . This paper presents some extremal results about the spectral radius ρ α ( G ) of the matrix A α ( G ) . In particular, we give a lower bound on the spectral radius ρ α ( G ) in terms of order and independence number. In addition, we obtain an upper bound for the spectral radius ρ α ( G ) in terms of order and minimal degree. Furthermore, for n > l > 0 and 1 ≤ p ≤ (cid:98) n − l 2 (cid:99) , let G p ∼ = K l ∨ ( K p ∪ K n − p − l ) be the graph obtained from the graphs K l and K p ∪ K n − p − l and edges connecting each vertex of K l with every vertex of K p ∪ K n − p − l . We prove that ρ α ( G p + 1 ) < ρ α ( G p ) for 1 ≤ p ≤ (cid:98) n − l 2 (cid:99) − 1.


Introduction
Let G = (V(G), E(G)) be a simple and undirected connected graph with vertex set V(G) = {v 1 , v 2 , . . . , v n } and edge set E(G) = {e 1 , e 2 , . . . , e m }. To simplify the notation, we represent a vertex just by u and then an edge is just represented by uv, we say that u is adjacent to v, or that u and v are neighbors and we write u ∼ v. The adjacency matrix of G, denoted by A(G) = (a ij ) is a symmetric matrix of order n such that a ij = 1 if v i and v j are adjacent, and 0 otherwise. Let d i = d v i = d G (v i ) be the degree of vertex v i in G. The largest and smallest vertex degrees of G are denoted by ∆(G) and δ(G), respectively. The degree matrix of G, denoted by D(G), is the diagonal matrix with diagonal entries the vertex degrees of G. A k-regular graph is a graph where every vertex has degree k. The complement of a graph G is represented by G. The spectrum of a matrix M, will be denoted by Sp(M). In this paper, the complete graph of order n is denoted by K n . An independent set is a set of vertices in a graph, where no two vertices are adjacent. The independence number of a graph G is the number of vertices of the largest independent set in G, denoted by γ(G) or just γ if there is no ambiguity.The line graph of a graph G, denoted by L G , is the graph whose vertex set is the edge set of G, where two vertices of L G are adjacent, if and only if, the corresponding edges are incident in G. The signless Laplacian matrix and Laplacian matrix of G are defined as Q(G) = D(G) + A(G) and L(G) = D(G) − A(G), respectively. We denote by µ 1 the Laplacian spectral radius of G, by q 1 the signless Laplacian spectral radius of G, and by λ 1 the adjacency spectral radius or spectral radius of G. V. Nikiforov in Reference [1], define the matrix A α (G) = αD (G) + (1 − α) A (G) , with 0 ≤ α ≤ 1.

Its straightforward verified that
A 0 (G) = A(G), 2A 1 2 (G) = Q(G), A 1 = D(G), and We denote by ρ α (G) to the spectral radius of A α (G) or well called the α-index of G. The join of two vertex disjoint graphs G 1 and G 2 is the graph obtained from the disjoint union G 1 ∪ G 2 by adding new edges from each vertex in G 1 to every vertex in G 2 . It is usually denoted by G 1 ∨ G 2 . This graph operation can be generalized in the following way: Let H be a graph of order k and V(H) = {1, 2, . . . , k}. Let F = {G 1 , G 2 , . . . , G k } be a set of pairwise vertex disjoint graphs. Here, each vertex j ∈ V(H) is assigned to the graph G j ∈ F . Let G be the graph obtained from the graphs G 1 , G 2 , . . . , G k and the edges connecting each vertex of G i with all the vertices of G j , for all edge ij ∈ E(H). That is, G is the graph with vertex set This graph is designated the H − join (or generalized composition) of the graphs G 1 , G 2 , . . . , G k [2][3][4], and it is denoted by In Reference [5], the authors determine the unique graph with maximal α-index among all connected graphs with diameter d, and determine the unique graph with minimal α-index among all connected graphs with given clique number. In Reference [6], the extremal graph with maximal α-index with fixed order and cut vertices and the extremal tree which attains the maximal α-index with fixed order and matching number are characterized. In Reference [7], the authors obtain the extremal graphs with maximal α-index with fixed order and diameter at least k. In References [8,9], are characterized the graphs which have the minimal spectral radius among all the connected graphs of order n and some values of the independence number γ. In Reference [10], Nikiforov et al. shown that if T ∆ is a tree of maximal degree ∆, then the spectral radius of A α (T ∆ ) satisfies the tight inequality In Reference [11], the authors obtain the following sharp upper bound for the spectral radius of G.

Theorem 1 ([11]
). Let G be a simple graph with n vertices and m edges. Let δ = δ(G) be the minimum degree of vertices of G and λ 1 (G) be the spectral radius of the adjacency matrix of G. Then, Equality holds, if and only if, G is either a regular graph or a bidegreed graph where each vertex has degree either δ or n − 1.
In Reference [12], is presented a sharp lower bound on the signless Laplacian spectral radius of a graph in terms of independence number. Theorem 2 ([12]). Let G be a graph of order n and γ(G) = γ. Then with equality, if and only if, n = kγ and G ∼ = γK k .
This paper is organized in the following way. In Section 1, besides the main concepts and notation used throughout the paper we present some recent work that motivated the authors. In Section 2, for n = tγ + s where 0 ≤ s < γ and 0 ≤ α < 1, a lower bound for the α-index of graphs of order n and independence number γ is given, further it is shown equality cases.
In Section 3, for 0 ≤ α < 1, an upper bound for the α-index of graphs of order n, edge number m and minimal degree δ is given, we demonstrate that the equality holds, if and only if, is either G a regular graph or a bidegreed graph where each vertex is of degree either δ or n − 1. In Section 4, for 0 ≤ α < 1, we present an ordering of α-index in the class of the graphs K l ∨ (K p ∪ K n−p−l ) where 1 ≤ p ≤ n−l 2 .

Spectral Radius and Independence Number
In this section, we present a lower bound for the spectral radius of graphs in terms of the order and independence number which generalizes and improves the lower bound presented in Theorem 2. We first present some lemmas used in the proof of our result. For general properties of the matrices A α (G) we refer the reader to Reference [1]. In particular, we frequently use the facts that ρ α (G) is non-decreasing in α ([1], Proposition 4).

Lemma 1 ([1]
). Let G be a graph of order n and H be any subgraph of G. If 0 ≤ α ≤ 1 then

Remark 1 ([12]
). If S ⊆ V(G) is a maximal independent set of G, then for each u ∈ V(G) − S, there exists some vertex v ∈ S such that uv ∈ E(G). Moreover, for any w ∈ V(G) − (S ∪ {u}) with uw / ∈ E(G) and wv ∈ E(G), there exists some vertex v 1 ∈ S − v such that uv 1 ∈ E(G) or wv 1 ∈ E(G).

Remark 2 ([12]
). For a graph G on n vertices non-isomorphic to the graph K n , we have

Lemma 2 ([13]
). Let G be a graph of order n and m edges. Let 0 ≤ α < 1. Then, with equality, if and only if, G is a regular graph.

Lemma 3.
Let G be a graph of order n and independence number γ with n = tγ + s where 0 ≤ s < γ and 1 < γ < n. Then G ∼ = sK t+1 ∪ (γ − s)K t , if and only if, in G the following conditions are verified (a) there exists t maximal independent sets S 1 , . . . , S t and an independent set S t+1 of cardinality s where S i ∩ S j = φ for i, j = 1, . . . , t + 1, i = j; (b) each vertex of S i is adjacent to a single vertex of S j for i, j = 1, . . . , t + 1, i = j.
Proof. Let G be a graph of order n and independence number γ that verifies the conditions (a) and (b). If t = 1 and s > 0 or t = 2 and s = 0 then G ∼ = sK t+1 ∪ (γ − s)K t , the result holds. Otherwise, let x ∈ S k , y ∈ S l and z ∈ S m where k = 1, . . . , t + 1; l, m = 1, . . . , t and k = m = l = k. Suppose xy ∈ E(G) and w} would be an independent set in G of cardinality γ + 1 which is a contradiction. By repeated applications of the above argument we can conclude that be a graph of order n and independence number γ with n = tγ + s where 0 ≤ s < γ and 1 < γ < n, then G has a proper subgraph H of order n and independence number γ.
Proof. Suppose G ∼ = sK t+1 ∪ (γ − s)K t then G does not verify some of the conditions (a) or (b) of Lemma 3. Suppose G does not verify the condition (a), then the following two cases may occur: disjoint independent sets which would be a contradiction). Then, by Remark 2, we can constructed S i ∼ = K s then we can constructed a new graph H 2 ∼ = G − e on n vertices and independence number γ, where e ∈ E(G 1 ), the result holds. Now, suppose that G verify condition (a) but does not verify condition (b), this is, we assume that G has t maximal independent sets S 1 , . . . , S t and an independent set S t+1 of cardinality s such that S i ∩ S j = φ for i, j = 1, . . . , t + 1 with i = j. As G does not check condition (b) there exists v ∈ S i and u, w ∈ S j for some i, j = 1, . . . , t + 1, i = j and u = w such that uv ∈ E(G) and wv ∈ E(G).
Then, we constructed a new graph H 3 ∼ = G − uv on n vertices and independence number γ, the result holds. (ii) Suppose that each vertex of S i is adjacent to a single vertex of S j for i = 1, . . . , t and j = 1, . . . , t + 1, i = j, then we use the same techniques applied in the proof from Lemma 3, we can see that G − S t+1 ∼ = γK t . Now, let i = t + 1 and G − S t+1 ∼ = γK t . We claim, v is adjacent to all the vertices of a connected component of G − S t+1 isomorphic to K t . Otherwise, there exists v m in each connected component of G − S t+1 isomorphic to K t such that vv m / ∈ E(G) for m = 1, . . . , γ. So, S = {v, v 1 , . . . , v γ } would be an independent set in G of cardinality γ + 1 which is a contradiction. Then, there exists Z = {v 1 , . . . , v t } such that v p ∈ V(G) − S t+1 , vv p ∈ E(G) for p = 1, . . . , t and v p v q ∈ E(G) for p, q = 1, . . . , t. Thus, u / ∈ Z or w / ∈ Z. If u / ∈ Z then we constructed a new graph H 4 ∼ = G − uv of order n and independence number γ.
The proof is complete.
Let G and H be two graphs of order n, we will say that G and H are comparable, if and only if, H is subgraph of G or G is subgraph of H.

Lemma 5.
Let G be a graph of order n and independence number γ with n = tγ + s where 0 ≤ s < γ and 1 < γ < n, then sK t+1 ∪ (γ − s)K t is a subgraph of G.
Proof. Let G 1 ∼ = sK t+1 ∪ (γ − s)K t then by Lemma 3 in G 1 the following conditions are verified (a) there exists t maximal independent sets S 1 , . . . , S t and an independent set S t+1 of cardinality s where S i ∩ S j = φ for i, j = 1, . . . , t + 1, i = j and (b) each vertex of S i is adjacent to a single vertex of S j for i, j = 1, . . . , t + 1, i = j. Let G 2 ∼ = G 1 − uv where u ∈ S i , v ∈ S j and uv ∈ E(G 1 ) for some i, j = 1, . . . , t + 1, i = j. Thus, in G 2 some of the sets S i ∪ {v} or S j ∪ {u} is an independent set of cardinality γ + 1. Since G is a graph of order n and independence number γ then G is not a proper subgraph of G 1 . Now, suppose G and G 1 are not comparable graphs. By Lemma 4, G has a proper subgraph H 1 of order n and independence number γ. Clearly, H 1 ∼ = G 1 . Then by Lemma 4, H 1 has a proper subgraph H 2 of order n and independence number γ. Clearly, H 2 ∼ = G 1 . By repeated applications of previous argument, we can conclude K n is a graph of orden n and independence number γ where 1 < γ < n which is a contradiction. Hence, G and G 1 are comparable graphs. Since G is not a proper subgraph of G 1 then G 1 is a subgraph of G. The proof is complete.
Below we present the main result of the section which is a lower bound for the spectral radius of graphs in terms of order and independence number.

Theorem 3. Let G be a graph of order n and independence number γ with n
• If s = 0, then the equality holds, if and only if, G ∼ = γK t .
Proof. Let G be a graph of order n and independence number γ with n = tγ + s where 0 ≤ s < γ. First consider γ = 1 then s = 0 and n = t. By Remark 1, G ∼ = K n ∼ = γK t the result holds. Now, let γ = n then s = 0 and t = 1. Then, G ∼ = K n ∼ = γK t the result is true. Now, we assume 1 < γ < n. By Lemma 5, Suppose the equality holds, s = 0 and G ∼ = G * then the edge number m of G is greater than γt(t − 1) 2 .
which is a contradiction. The proof is complete.
The following result which is a direct consequence of Theorem 3, presents a lower bound for the signless Laplacian spectral radius and this in turn improves the bound of Theorem 2. Corollary 1. Let G be a graph of order n and independence number γ with n = tγ + s where 0 ≤ s < γ, then • If s = 0, then the equality holds, if and only if, Proof. Taking α = 1 2 then 2A α (G) = Q (G), by Theorem 3 the result holds.

Spectral Radius and Minimal Degree
In this section, we provide the necessary definitions and lemmas on which our main results rely. We begin with some simple matrix results. Let B = (b ij ) be an m × n matrix. Then s i (B) will denote the i-th row sum of B, that is, An immediate consequence is the following result.

Lemma 7.
Let G be a simple connected graph of order n with A α (G) = A α where 0 ≤ α < 1. Let P be any polynomial and R v (P(A α )) be the row sum of P(A α ) corresponding to v ∈ V(G), then Both equalities above holds, if and only if, the row sums of P(A α ) are all equal.
As a consequence of Lemma 7, we obtain the following result which generalize Theorem 1.

Theorem 4.
Let G be a simple connected graph of order n and minimal degree δ with m edges and 0 ≤ α < 1. Then The equality holds, if and only if, G is either a regular graph or a bidegreed graph where each vertex has degree either δ or n − 1.
Proof. Let G be a graph of order n, minimal degree δ and m edges and R v (A α ) be the row sum of A α corresponding to v ∈ V(G) where A α denote the matrix A α (G). We can easily see that Thus, Thereby, For the linearity, Taking In particular, Solving the inequality (2) for ρ α (G), we obtain Suppose the equality in (3) holds, then all the inequalities in the above argument must be equalities. This implies that for all v ∈ V(G). Now, suppose u 1 and v 1 are two non-adjacent vertices in G then d v 1 < n − 1. By (4), we have d u 1 = δ. Analogously, we can prove that d v 1 = δ. Hence, the affirmation "Any pair of non-adjacent vertices in G have degree δ" is true. Now, if there exists v ∈ V(G) such that d v > δ then by above affirmation, we conclude v is adjacent to u for all u ∈ V(G) with u = v, this is, d v = n − 1 which implies either G is a regular graph of degree δ or G is a bidegreed graph where each vertex has degree either δ or n − 1. Conversely, if G is a regular graph then the equality holds. Now, suppose G is a bidegreed graph where each vertex has degree δ or n − 1. (1) and (4) we have Hence, The proof is complete.

Corollary 2.
Let G be a graph of order n and minimal degree δ with m edges then Equality holds, if and only if, G is either a regular graph or a bidegreed graph where each vertex has degree either δ or n − 1.
Lemma 8 ([15,16]). Let G be a graph of order n with m ≥ 1 edges. Let q i be the i-th greatest signless Laplacian eigenvalue of G and λ i (L G ) be the i-th greatest eigenvalue of the line graph of G. Then for i = 1, 2, . . . , k, where k = min{n, m}. In addition, if m > n, then λ i (L G ) = −2 for i ≥ n + 1 and if n > m, then q i = 0 for i ≥ m + 1.
As a direct consequence of Corollary 2 and Lemma 8, we obtain the following result.
Corollary 3. Let G be a graph of order n and minimal degree δ with m edges then Equality holds, if and only if, G is either a regular graph or a bidegreed graph where each vertex has degree δ or n − 1.

Monotonicity of the α-Indices of Graphs with 0 ≤ α < 1
In this section, for 0 ≤ α < 1, we present an ordering of α-indices in the class of the graphs G p ∼ =K l ∨ (K p ∪ K n−p−l ) where 1 ≤ p ≤ n−l 2 . The graphs G p plays an important role in the representation of graphs that relate the join operation between an arbitrary family of graphs, in particular maximize some topological indices of graphs in terms of edge connectivity and vertices connectivity such as Energy, Estrada index, Spread (see References [17][18][19]). Furthermore, in graphs of communication or transportation networks, the edge connectivity is an important measure of reliability, the study of the line graph of these graphs is relevant. In Theorem 8 [20], for 1 ≤ p ≤ n−l 2 is presented an ordering between the spectral radii of the distance matrices D α (G p ) where α ∈ (α, 1), being α = max{α p : 2 ≤ p ≤ n−l 2 } and α p the unique zero of the function e(x, α) = (4α − 3)x + α 2 (2l − 4n) + α(3n − 3l + 4) + l − 3 in the interval 3 4 , 3n−l 4n−2l . In this section, we prove that this same order is conserved for the spectral radii of the matrices A α (G p ) with α ∈ [0, 1). We begin presenting a result for the spectrum of the matrices A α (G p ) with α ∈ [0, 1). In Theorem 5 [2], the spectrum of the adjacency matrix of the H-join of regular graphs is obtained. The version of this result for the matrices A α with 0 ≤ α < 1 is given below, its proof is similar.
where H is a graph of order k and G j is a r j -regular graph of order n j for j = 1, . . . , k. Then, the spectrum of the matrix A α (G) with 0 ≤ α < 1 is where M(G) is the matrix of order k given by Let l and n be fixed positive integer and p be a positive integer such that 1 ≤ p ≤ n−l 2 , by (5) we have We remember that the eigenvalues of A α (K s ) are s − 1 and αs − 1 with multiplicity s − 1. The following result is a direct consequence of Theorem 5. Hence, We claim ρ α (G p ) > ρ α (G p+1 ). Otherwise, if ρ α (G p ) ≤ ρ α (G p+1 ) then ρ α (G p+1 ) ≥η j , for all j. Taking x = ρ α (G p+1 ) in (6), we obtain which is a contradiction. Then, we conclude ρ α (G p ) > ρ α (G p+1 ) for all 1 ≤ p ≤ n−l 2 − 1.