ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P1.06
https://doi.org/10.26493/1855-3974.2697.43a
(Also available at http://amc-journal.eu)
On the Aα-spectral radius of connected graphs*
Abdollah Alhevaz , Maryam Baghipur
Faculty of Mathematical Sciences, Shahrood University of Technology,
P.O. Box: 316-3619995161, Shahrood, Iran
Hilal Ahmad Ganie
Department of School Education, JK Govt. Kashmir, India
Kinkar Chandra Das †
Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea
Received 14 September 2021, accepted 23 June 2022, published online 5 October 2022
Abstract
For a simple graph G, the generalized adjacency matrix Aα(G) is defined as Aα(G) =
αD(G) + (1 − α)A(G), α ∈ [0, 1], where A(G) is the adjacency matrix and D(G) is the
diagonal matrix of the vertex degrees. It is clear that A0(G) = A(G) and 2A 1
2
(G) =
Q(G) implying that the matrix Aα(G) is a generalization of the adjacency matrix and the
signless Laplacian matrix. In this paper, we obtain some new upper and lower bounds for
the generalized adjacency spectral radius λ(Aα(G)), in terms of vertex degrees, average
vertex 2-degrees, the order, the size, etc. The extremal graphs attaining these bounds are
characterized. We will show that our bounds are better than some of the already known
bounds for some classes of graphs. We derive a general upper bound for λ(Aα(G)), in
terms of vertex degrees and positive real numbers bi. As application, we obtain some new
upper bounds for λ(Aα(G)). Further, we obtain some relations between clique number
ω(G), independence number γ(G) and the generalized adjacency eigenvalues of a graph
G.
Keywords: Adjacency matrix, signless Laplacian matrix, generalized adjacency matrix, spectral ra-
dius, degree sequence, clique number, independence number.
Math. Subj. Class. (2020): Primary: 05C50, 05C12; Secondary: 15A18.
*The authors would like to thank the handling editor and two anonymous referees for their detailed constructive
comments that helped improve the quality of the paper.
†Corresponding author. Partially supported by the National Research Foundation of the Korean government
with grant No. 2021R1F1A1050646.
E-mail addresses: a.alhevaz@shahroodut.ac.ir (Abdollah Alhevaz), maryamb8989@gmail.com (Maryam
Baghipur), hilahmad1119kt@gmail.com (Hilal Ahmad Ganie), kinkardas2003@gmail.com (Kinkar Chandra
Das)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Ars Math. Contemp. 23 (2023) #P1.06
1 Introduction
Let G = (V (G), E(G)) be a simple connected graph with vertex set V (G) = {v1, v2, . . . ,
vn} and edge set E(G). The order of G is the number n = |V (G)| and its size is the
number m = |E(G)|. The set of vertices adjacent to v ∈ V (G), denoted by N(v), refers
to the neighborhood of v. The degree of v, denoted by dG(v) (we simply write dv if it
is clear from the context) means the cardinality of N(v). A graph is called regular if all
vertices have the same degree. The graph G is the complement of the graph G. Moreover,
the complete graph Kn, the complete bipartite graph Ks,t, the path Pn, the cycle Cn and
the star Sn are defined in the conventional way. The distance between two vertices u, v ∈
V (G), denoted by duv , is defined as the length of a shortest path between u and v in G.
The diameter of G is the maximum distance between any two vertices of G. Let mi be the
average degree of the adjacent vertices of vertex vi in G. If vi is an isolated vertex in G,
then we assume that mi = 0. Hence we can write
mi =
0 di = 0.1
di
∑
j:j∼i dj otherwise.
Let pi be the average degree of the vertices non-adjacent to vertex vi in G. If vi is adjacent
to all the remaining vertices, then we assume that pi = 0. Then we can write
pi =
0 di = n− 1.∑j:j≁i, j ̸=i dj
n−di−1 otherwise.
Let D(G) be the diagonal matrix of vertex degrees and A(G) be the adjacency matrix
of G. The signless Laplacian matrix of G is Q(G) = D(G) + A(G). Its eigenvalues can
be arranged as: q1(G) ≥ q2(G) ≥ · · · ≥ qn(G). In [20], Nikiforov proposed the following
matrix:
Aα(G) = αD(G) + (1− α)A(G), 0 ≤ α ≤ 1,
calling it the generalized adjacency matrix of G. Obviously, A0(G) = A(G), 2A 1
2
(G) =
Q(G), A1(G) = D(G) and Aα(G) − Aβ(G) = (α − β)L(G), where L(G) is the
well-studied Laplacian matrix of G, defined as L(G) = D(G) − A(G). Therefore, the
family Aα(G) can extend both A(G) and Q(G). The matrix Aα(G) is a real symmetric
matrix, therefore we can arrange its eigenvalues as λ1(Aα(G)) ≥ λ2(Aα(G)) ≥ · · · ≥
λn(Aα(G)), where λ1(Aα(G)) is called the generalized adjacency spectral radius of G.
Afterwards, we will denote λ1(Aα(G)) by λ(Aα(G)). If G is a connected graph and
α ̸= 1, then the matrix Aα(G) is non-negative and irreducible. Therefore by the Perron-
Frobenius theorem, λ(Aα(G)) is the simple eigenvalue and there is a unique positive unit
eigenvector x corresponding to λ(Aα(G)), which is called the generalized adjacency Per-
ron vector of G.
A column vector x = (x1, x2, . . . , xn)T ∈ Rn can be considered as a function defined
on V (G) which maps vertex vi to xi, i.e., x(vi) = xi for i = 1, 2, . . . , n. Then,
⟨x, Aαx⟩ = xTAα(G)x = α
n∑
i=1
dix
2
i + 2(1− α)
∑
i∼j
xixj ,
A. Alhevaz et al.: On the Aα-spectral radius of connected graphs 3
and λ is an eigenvalue of Aα(G) corresponding to the eigenvector x if and only if x ̸= 0
and
λxi = αdixi + (1− α)
∑
j∼i
xj , i = 1, 2, . . . , n.
These equations are called the (λ, x)-eigenequations of G. For a normalized column vector
x ∈ Rn, by the Rayleigh’s principle, we have
λ(Aα(G)) ≥ xTAα(G)x
with equality if and only if x is the generalized adjacency Perron vector of G.
The research on the (adjacency, signless Laplacian) spectrum is an intriguing topic
during past two decades [4, 10, 22]. At the same time, the adjacency or signless Laplacian
spectral radius have attracted many interests among the mathematical literature including
linear algebra and graph theory. An interesting problem in the spectral graph theory is
to obtain bounds for the (adjacency, signless Laplacian) spectral radius connecting it with
different parameters associated with the graph. Another interesting problem which is worth
to mention is to characterize the extremal graphs for the (adjacency, signless Laplacian)
spectral radius among all graphs of order n or among a special class of graphs of order
n. The spectral radius λ(G) of the adjacency matrix A(G), called the spectral radius (or
adjacency spectral radius) of the graph G and the spectral radius q1(G) of the signless
Laplacian matrix Q(G), called signless Laplacian spectral radius of the graph G, are both
well studied and their spectral theories are well developed. Various papers can be found in
the literature regarding the establishment of bounds for λ(G) and q1(G) connecting them
with different parameters associated with the structure of the graph G. Since the matrix
Aα(G) is a generalization of the matrices A(G) and Q(G), therefore it will be interesting
to see whether the results which already hold for the spectral radius of the matrices A(G)
and/or Q(G) can be extended to the spectral radius of the Aα(G). This is one of the
motivation to study the spectral radius of the matrix Aα(G).
Let A(G) be the adjacency matrix of the graph G and let B be a real diagonal matrix
of order n. In 2002, Bapat et al. [1] defined the matrix L
′
= B − A(G) and called it
the perturbed Laplacian matrix of the graph G. The aim of introducing this matrix was to
generalize the results that hold for the adjacency matrix and the Laplacian matrix L(G) of
the graph to some general class of matrices. For α ̸= 1, it is easy to see that
Aα(G) = αD(G) + (1− α)A(G) = (α− 1)
( α
α− 1
D(G)−A(G)
)
.
Clearly αα−1D(G) is a diagonal matrix with real entries, giving that the matrix Aα(G) is
a scaler multiple of a perturbed Laplacian matrix. This is another motivation to study the
spectral properties of the matrix Aα(G).
Although the generalized adjacency matrix Aα(G) of a graph G was introduced in
2017, but a large number of papers can be found in the literature regarding the spectral
properties of this matrix. Like other graph matrices, most of these papers are regarding the
generalized spectral radius λ(Aα(G)). In fact, various upper and lower bounds connecting
λ(Aα(G)) with different graph parameters and the graphs attaining these bounds can be
found in the literature. For some recent works regarding the spectral properties of Aα(G),
we refer to [8, 9, 11, 13, 14, 15, 16, 17, 21, 23, 24].
The rest of this paper is organized as follows. In Section 2, we obtain some new upper
and lower bounds for λ(Aα(G)), in terms of vertex degrees, average vertex 2-degrees, the
4 Ars Math. Contemp. 23 (2023) #P1.06
order, the size, etc. The extremal graphs attaining these bounds are characterized. We will
show that our bounds are better than some of the already known bounds for some classes
of graphs. In Section 3, we derive a general upper bound for λ(Aα(G)), in terms of vertex
degrees and positive real numbers bi. As application, we obtain some new upper bounds
for λ(Aα(G)). In Section 4, we obtain some relations between clique number ω(G), inde-
pendence number γ(G) and the generalized adjacency eigenvalues. We conclude this paper
by a remark in Section 5.
2 Bounds on generalized adjacency spectral radius
The average 2-degree of a vertex vi ∈ V (G) is denoted by mi = m(vi) and is defined as
mi =
∑
k:k∼i
dk
di
, where dk is the degree of the vertex vk.
The following gives an upper bound for the generalized adjacency spectral radius
λ(Aα(G)) of a graph in terms of the vertex degrees, the average vertex 2-degrees and
the parameter α.
Theorem 2.1. Let G be a graph of order n having vertex degrees di, vertex average 2-
degrees mi, 1 ≤ i ≤ n, and let α ∈ [0, 1]. Then
λ(Aα(G)) ≤ max
1≤i≤n
{
αdi + (1− α)
√
dimi
}
.
Moreover, the equality holds if G is a k-regular graph.
Proof. Let x = (x1, . . . , xn) be the generalized adjacency Perron vector of G and let
∥x∥ = 1. For any vi ∈ V (G), we have λ(Aα(G))xi = αdixi+(1−α)
∑
j:j∼i xj . Hence
λ2(Aα(G))x
2
i = α
2d2ix
2
i + 2α(1− α)dixi
∑
j:j∼i
xj + (1− α)2
∑
j:j∼i
xj
2 . (2.1)
By Cauchy-Schwarz inequality, we obtain∑
j:j∼i
xj
2 ≤ di ∑
j:j∼i
x2j . (2.2)
Therefore from (2.1) and (2.2), we get
λ2(Aα(G))x
2
i ≤ α2d2ix2i + 2αdixi
[
λ(Aα(G))xi − αdixi
]
+ (1− α)2di
∑
j:j∼i
x2j .
Thus, taking sum over all vi ∈ V (G), we get∑
vi∈V (G)
λ2(Aα(G))x
2
i
≤
∑
vi∈V (G)
[
2αdiλ(Aα(G))− α2d2i
]
x2i + (1− α)2
∑
vi∈V (G)
di
∑
j:j∼i
x2j
=
∑
vi∈V (G)
[
2αdiλ(Aα(G))− α2d2i
]
x2i + (1− α)2
∑
vi∈V (G)
dimix
2
i
=
∑
vi∈V (G)
[
2αdiλ(Aα(G))− α2d2i + (1− α)2dimi
]
x2i
A. Alhevaz et al.: On the Aα-spectral radius of connected graphs 5
as ∑
vi∈V (G)
di
∑
j:j∼i
x2j =
∑
vi∈V (G)
x2i
∑
j:j∼i
dj =
∑
vi∈V (G)
dimix
2
i .
From the above result, we obtain∑
vi∈V (G)
(
λ2(Aα(G))− 2αdiλ(Aα(G)) + α2d2i − (1− α)2dimi
)
x2i ≤ 0.
This is only true if there exist a vertex, say vj ∈ V (G), such that
λ2(Aα(G))− 2αdjλ(Aα(G)) + α2d2j − (1− α)2djmj ≤ 0,
hence, we get
λ(Aα(G)) ≤ αdj + (1− α)
√
djmj ≤ max
1≤i≤n
{
αdi + (1− α)
√
dimi
}
.
Now, suppose that G is a k-regular graph. So, for i = 1, . . . , n, we have di = mi = k,
then αdi + (1− α)
√
dimi = k and λ(Aα(G)) = k. This shows that equality occurs for a
regular graph.
For α = 0, the upper bound given by Theorem 2.1 reduces to the upper bound in the
following corollary.
Corollary 2.2 ([5]). Let G be a graph of order n having vertex degrees di, vertex average
2-degrees mi, 1 ≤ i ≤ n. Then
λ(A(G)) ≤ max
1≤i≤n
√
di mi.
The following upper bound for the generalized adjacency spectral radius λ(Aα(G)), in
terms of vertex degrees and average vertex 2-degrees was obtained in [20]:
Theorem 2.3. If G is a graph with no isolated vertices, then
λ(Aα(G)) ≤ max
vj∈V
{
αdj + (1− α)mj
}
.
If α ∈
(
1
2 , 1
)
and G is connected, equality holds if and only if G is regular.
Remark 2.4. For non-regular graphs the upper bound given by Theorem 2.1 and the upper
bound given by Theorem 2.3 are incomparable for different values of α. For example,
consider the graph G = K4 − e. For this graph we have d1 = 2, d2 = 3, d3 = 2, d4 =
3,m1 = 3,m2 =
7
3 ,m3 = 3 and m4 =
7
3 . By Theorem 2.3, we have
λ(Aα(G)) ≤ max
{
3− α, 7
3
+
2
3
α
}
.
It is easy to see that
max
{
3− α, 7
3
+
2
3
α
}
=
{
7
3 +
2
3α for α > 0.4,
3− α for α ≤ 0.4.
6 Ars Math. Contemp. 23 (2023) #P1.06
Also, by Theorem 2.1, we have
λ(Aα(G)) ≤ max
{√
6 + (2−
√
6)α,
√
7 + (3−
√
7)α
}
=
√
7 + (3−
√
7)α.
For α ≤ 0.4, we have 3 − α >
√
7 + (3 −
√
7)α giving that α < 5−
√
7
9 ≈ 0.2615. This
gives that for 0 ≤ α < 5−
√
7
9 , the upper bound given by Theorem 2.1 is better than the
upper bound given by Theorem 2.3; while as for 5−
√
7
9 ≤ α ≤ 0.4, the upper bound given
by Theorem 2.3 is better than the upper bound given by Theorem 2.1 for the graph K4 − e.
For the graph G = K1,3, we have d1 = 3, d2 = 1, d3 = 1, d4 = 1,m1 = 1,m2 = 3,m3 =
3 and m4 = 3. By Theorem 2.3, we have
λ(Aα(G)) ≤ max
{
1 + 2α, 3− 2α
}
.
It is easy to see that
max
{
1 + 2α, 3− 2α
}
=
{
3− 2α for α < 0.5,
1 + 2α for α ≥ 0.5.
Also, by Theorem 2.1, we have
λ(Aα(G)) ≤ max
{√
3 + (3−
√
3)α,
√
3 + (1−
√
3)α
}
=
√
3 + (3−
√
3)α.
For α < 0.5, we have 3− 2α >
√
3 + (3−
√
3)α giving that α < 3−
√
3
5−
√
3
≈ 0.38799. This
gives that for 0 ≤ α < 3−
√
3
5−
√
3
, the upper bound given by Theorem 2.1 is better than the
upper bound given by Theorem 2.3; while as for 3−
√
3
5−
√
3
≤ α < 0.5, the upper bound given
by Theorem 2.3 is better than the upper bound given by Theorem 2.1 for the graph K1,3.
The following gives another upper bound for the generalized adjacency spectral radius
λ(Aα(G)) of a graph G in terms of the vertex degrees, the average vertex 2-degrees and
the unknown parameter β.
Theorem 2.5. Let G be a connected graph of order n having vertex degrees di, average
vertex 2-degrees mi, 1 ≤ i ≤ n, and let α ∈ [0, 1). Then
λ(Aα(G)) ≤ max
1≤i≤n
{
−β +
√
β2 + 4di(αdi + (1− α)mi + β)
2
}
, (2.3)
where β ≥ 0 is an unknown parameter. Equality occurs if and only if G is a regular graph.
Proof. Let x = (x1, . . . , xn) be the generalized adjacency Perron vector of G and let
xi = max
1≤j≤n
xj .
Since
λ2(Aα(G))x = (Aα(G))
2
x = (αD + (1− α)A)2x
= α2D2x+ α(1− α)DAx+ α(1− α)ADx+ (1− α)2A2x,
A. Alhevaz et al.: On the Aα-spectral radius of connected graphs 7
we have
λ2(Aα(G))xi = α
2d2ixi + α(1− α)di
∑
j:j∼i
xj + α(1− α)
∑
j:j∼i
djxj
+ (1− α)2
∑
j:j∼i
∑
k:k∼j
xk.
Now, we consider a simple quadratic function of λ(Aα(G)):(
λ2(Aα(G)) + βλ(Aα(G))
)
x =
(
α2D2x+ α(1− α)DAx+ α(1− α)ADx
+ (1− α)2A2x
)
+ β(αDx+ (1− α)Ax).
Considering the i-th equation, we have(
λ2(Aα(G)) + βλ(Aα(G))
)
xi = α
2d2ixi + α(1− α)di
∑
j:j∼i
xj + α(1− α)
∑
j:j∼i
djxj
+ (1− α)2
∑
j:j∼i
∑
k:k∼j
xk + β
(
αdixi + (1− α)
∑
j:j∼i
xj
)
.
One can easily see that
α(1− α)di
∑
j:j∼i
xj ≤ α(1− α)d2ixi, α(1− α)
∑
j:j∼i
djxj ≤ α(1− α)dimixi,
(1− α)2
∑
j:j∼i
∑
k:k∼j
xk ≤ (1− α)2dimixi, (1− α)
∑
j:j∼i
xj ≤ (1− α)dixi.
Hence, we obtain(
λ2(Aα(G)) + βλ(Aα(G))
)
xi ≤ di(αdi + (1− α)mi)xi + βdixi,
that is, λ2(Aα(G)) + βλ(Aα(G))− di(αdi + (1− α)mi + β) ≤ 0,
that is, λ(Aα(G)) ≤
−β +
√
β2 + 4di(αdi + (1− α)mi + β)
2
.
From this the inequality (2.3) follows.
Suppose that equality occurs in (2.3). Then all the inequalities in the above argument
occur as equalities. Thus we obtain
α(1− α)di
∑
j:j∼i
xj = α(1− α)d2ixi, α(1− α)
∑
j:j∼i
djxj = α(1− α)dimixi,
(1− α)2
∑
j:j∼i
∑
k:k∼j
xk = (1− α)2dimixi, (1− α)
∑
j:j∼i
xj = (1− α)dixi.
Therefore we must have xj = xi for any j : j ∼ i and xk = xi for any k : k ∼ j, j ∼ i.
Let U = {vℓ : xℓ = xi}. Now we have to prove that U = V (G). Assume to the contrary
8 Ars Math. Contemp. 23 (2023) #P1.06
that U ̸= V (G). Then there exists a vertex r in U such that N(r) ⊆ U and t ∈ V (G)\U
with t ∼ s, where s ∈ N(r). Then xt < xi. One can easily see that
λ(Aα(G)) <
−β +
√
β2 + 4dr(αdr + (1− α)mr + β)
2
≤ max
1≤i≤n
{
−β +
√
β2 + 4di(αdi + (1− α)mi + β)
2
}
,
a contradiction as the equality holds in (2.3). Therefore U = V (G). Then x1 = x2 =
· · · = xn and λ(Aα(G)) = di, i = 1, 2, . . . , n. Hence G is a regular graph.
Conversely, let G be a r-regular graph. Then
λ(Aα(G)) = r = max
1≤i≤n
{
−β +
√
β2 + 4di(αdi + (1− α)mi + β)
2
}
.
This completes the proof of the theorem.
The following upper bound for the generalized adjacency spectral radius λ(Aα(G)), in
terms of vertex degrees and average vertex 2-degrees was obtained in [20]:
λ(Aα(G)) ≤ max
1≤i≤n
{√
αd2i + (1− α)wi
}
, (2.4)
where wi = dimi for i = 1, . . . , n. Also, equality holds if and only if αd2i + (1− α)wi is
same for all i.
Remark 2.6. For a connected graph G of order n, the upper bound given by Theorem 2.5
reduces to the upper bound given by (2.4) for β = 0. For β ̸= 0, the upper bound given by
Theorem 2.5 is incomparable with the upper bound given by (2.4). For example, consider
the graph G = K1,3. For this graph, the upper bound (2.4) gives
λ(Aα(G)) ≤ max
{√
3 + 6α,
√
3− 3α
}
=
√
3 + 6α.
While as the upper bound given by Theorem 2.5 gives
λ(Aα(G)) ≤max
{
−β +
√
β2 + 12β + 12α+ 12
2
,
−β +
√
β2 + 4β − 8α+ 12
2
}
=
−β +
√
β2 + 12β + 12α+ 12
2
.
Taking β = 1, we have
−β +
√
β2 + 12β + 12α+ 12
2
=
−1 +
√
25 + 12α
2
<
√
3 + 6α
giving that 3α2 − 8α + 2 < 0. This last inequality holds provided that α > 4−
√
10
3 ≈
0.279240. This shows that for β = 1, the upper bound given by Theorem 2.5 is better
than the upper bound given by (2.4) for α > 4−
√
10
3 . Taking β = 0.5, it can be seen that
the upper bound given by Theorem 2.5 is better than the upper bound given by (2.4) for
α > 0.177 and for β = 0.1, it can be seen that the upper bound given by Theorem 2.5 is
A. Alhevaz et al.: On the Aα-spectral radius of connected graphs 9
better than the upper bound given by (2.4) provided that α > 0.008.
Since for β = 0, the upper bounds given by Theorem 2.5 and inequality (2.4) are
same and for the graph K1,3, it follows from the above discussion that for small value of β
the upper bound given by Theorem 2.5 behaves well for all α, incomparable to the upper
bound given by (2.4). This gives that the choice of parameter β in the upper bound given
by Theorem 2.5 can be helpful to obtain a better upper bound.
Let xi = min{xj , j = 1, . . . , n} be the minimum among the entries of the generalized
distance Perron vector x = (x1, . . . , xn) of the graph G. Proceeding similar to Theorem
2.5, we obtain the following lower bound for λ(Aα(G)), in terms of the vertex degrees, the
average vertex 2-degrees and the unknown parameter β.
Theorem 2.7. Let G be a connected graph of order n having vertex degrees di, average
vertex 2-degrees mi, 1 ≤ i ≤ n, and let α ∈ [0, 1). Then
λ(Aα(G)) ≥ min
1≤i≤n
{
−β +
√
β2 + 4di(αdi + (1− α)mi + β)
2
}
,
where β ≥ 0 is an unknown parameter. Equality occurs if and only if G is a regular graph.
The following lower bound for the generalized adjacency spectral radius λ(Aα(G)), in
terms of vertex degrees and average vertex 2-degrees was obtained in [20]:
λ(Aα(G)) ≥ min
1≤i≤n
{√
αd2i + (1− α)wi
}
, (2.5)
where wi = dimi for i = 1, . . . , n. Equality occurs if and only if αd2i +(1−α)wi is same
for all i.
Remark 2.8. For a connected graph G of order n, the lower bound given by Theorem 2.7
reduces to the lower bound given by (2.5), for β = 0. For β ̸= 0, the lower bound given by
Theorem 2.7 is incomparable with the lower bound given by (2.5). For example, consider
the graph G = K1,3. For this graph, the lower bound (2.5) gives
λ(Aα(G)) ≥ min
{√
3 + 6α,
√
3− 3α
}
=
√
3− 3α.
While as the lower bound given by Theorem 2.7 gives
λ(Aα(G)) ≥ min
{
−β +
√
β2 + 12β + 12α+ 12
2
,
−β +
√
β2 + 4β − 8α+ 12
2
}
=
−β +
√
β2 + 4β − 8α+ 12
2
.
Taking β = 1, we have
−β +
√
β2 + 4β − 8α+ 12
2
=
−1 +
√
17− 8α
2
>
√
3− 3α
giving that 4α2 + 20α − 8 > 0. This last inequality holds provided that α >
√
33−5
2 ≈
0.372281. This shows that for β = 1, the lower bound given by Theorem 2.7 is better
than the lower bound given by (2.5) for α >
√
33−5
2 . Taking β = 0.1, it can be seen that
the lower bound given by Theorem 2.7 is better than the lower bound given by (2.5) for
10 Ars Math. Contemp. 23 (2023) #P1.06
α > 0.09 and for β = 0.01, it can be seen that the lower bound given by Theorem 2.7 is
better than the lower bound given by (2.5) provided that α > 0.023.
Again, it follows from the above discussion that for small value of β the lower bound
given by Theorem 2.7 behaves well for all α, incomparison to the lower bound given by
(2.5) for the graph K1,3. This gives that the choice of parameter β in the lower bound given
by Theorem 2.7 can be helpful to obtain a better lower bound.
We note that if, in particular we take the parameter β in Theorem 2.5/Theorem 2.7
equal to the vertex covering number, the edge covering number, the clique number, the
independence number, the domination number, the generalized adjacency rank, minimum
degree, maximum degree, etc., then Theorems 2.5/ Theorem 2.7 gives upper bound/lower
bound for λ(Aα(G)), in terms of the vertex covering number, the edge covering number,
the clique number, the independence number, the domination number, the generalized ad-
jacency rank, minimum degree, maximum degree, etc.
Let Sn be the class of graphs of order n with maximum degree n − 1. Clearly,
K1,n−1, Kn ∈ Sn. The following result gives an upper bound for maxvj∈V {dj +mj} in
terms of order n and size m.
Lemma 2.9 ([3]). Let G be a graph of order n with m edges. Then
max
1≤j≤n
{
dj +mj
}
≤ 2m
n− 1
+ n− 2, (2.6)
with equality if and only if G ∈ Sn or G ∼= Kn−1 ∪K1.
We now generalize the above result.
Theorem 2.10. Let G be a graph of order n with m edges and real numbers β, θ with
β ≥ θ > 0. Then
max
1≤j≤n
{
βdj + θmj
}
≤ 2mθ
n− 1
+ β (n− 1)− θ, (2.7)
with equality if and only if G ∈ Sn or G ∼= Kn−1 ∪K1 (β = θ).
Proof. If β = θ > 0, then by Lemma 2.9, we get the required result in (2.7). Moreover, the
equality holds if and only if G ∈ Sn or G ∼= Kn−1 ∪K1 (β = θ). Otherwise, β > θ > 0.
Let vi be the vertex in G such that
max
1≤j≤n
{
βdj + θmj
}
= βdi + θmi.
We have 2m = di + dimi + (n− di − 1) pi, where pi is the average of the degrees of the
vertices non-adjacent to vertex vi in G. We consider the following two cases:
Case 1: di = n− 1. One can easily see that
max
1≤j≤n
{
βdj + θmj
}
= βdi + θmi =
2mθ
n− 1
+ β (n− 1)− θ.
In this case G ∈ Sn.
Case 2: di ≤ n− 2. Now, to arrive at (2.7), we need to show that
βdi + θmi ≤
di + dimi + (n− di − 1)pi
n− 1
θ + β (n− 1)− θ,
A. Alhevaz et al.: On the Aα-spectral radius of connected graphs 11
that is,
(n− di − 1)
(
(n− 1)β + (pi − 1−mi) θ
)
≥ 0,
that is,
(n− 1)β + (pi − 1−mi) θ ≥ 0,
that is,
(n− 1)β − (∆− δ + 1) θ ≥ 0, (2.8)
as mi ≤ ∆ and pi ≥ δ. We consider the following two subcases:
Subcase 2.1: G is disconnected. Then ∆ ≤ n−2. From (2.8), we obtain (n−1) (β−θ) >
0, which is true always as β > θ > 0. This shows that the inequality (2.8) strictly holds in
this case.
Subcase 2.2: G is connected. In this case ∆− δ ≤ n− 2, again it follows from (2.8) that
(n−1) (β−θ) > 0, which is true always as β > θ > 0. This shows that the inequality (2.8)
strictly holds in this case as well.
As an immediate consequence of Theorem 2.10, we get the following corollary.
Corollary 2.11. Let G be a graph of order n with m edges and real number α ≥ 12 . Then
max
1≤j≤n
{
αdj + (1− α)mj
}
≤ 2m (1− α)
n− 1
+ αn− 1, (2.9)
with equality if and only if G ∈ Sn or G ∼= Kn−1 ∪K1 (α = 1/2).
Combining Theorem 2.3 with Corollary 2.11, we get the following result, which gives
an upper bound for the generalized adjacency spectral radius λ(Aα(G)), in terms of the
order n, the size m and the parameter α.
Theorem 2.12. Let G be a graph of order n with m edges, with no isolated vertices and
let α ∈
[
1
2 , 1
]
. Then
λ(Aα(G)) ≤
2m (1− α)
n− 1
+ αn− 1.
If α ∈
(
1
2 , 1
)
and G is connected, equality holds if and only if G = Kn.
Let Γ be the class of graphs G = (V,E) such that the maximum degree vertex (of
degree ∆) are adjacent to the vertices of degree ∆ and non-adjacent to the vertices of
degree δ. If m is the number of edges in G (∈ Γ), then
2m = ∆(∆+ 1) + (n−∆− 1) δ.
The following result gives an upper bound for di +mi in terms of the order n, the size m,
the maximum degree ∆ and the minimum degree δ.
Lemma 2.13 ([3]). Let G be a graph of order n with m edges having maximum degree ∆
and minimum degree δ. Then
di +mi ≤
2m
n− 1
+ ∆− δ + ∆
n− 1
[
n− 2− (∆− δ)
]
,
with equality if and only if G ∈ Sn or G ∈ Γ.
12 Ars Math. Contemp. 23 (2023) #P1.06
The following result gives an upper bound for maxvj∈V {β dj + θmj} in terms of the
order n, the size m, the maximum degree ∆, the minimum degree δ and the parameters β,
θ.
Theorem 2.14. Let G be a graph of order n with m edges and real numbers β, θ with
β ≥ θ > 0. Then
max
vj∈V
{βdj + θmj} ≤
2mθ
n− 1
+ θ (∆− δ) + ∆
n− 1
[
β (n− 1)− θ(∆− δ + 1)
]
(2.10)
with equality if and only if G ∈ Sn or G ∈ Γ.
Proof. Let vi be a vertex in G such that
max
vj∈V
{β dj + θmj} = β di + θmi.
First we assume that β = θ. Then by Lemma 2.13, we obtain
max
vj∈V
{βdj + θmj} = β (di +mi) ≤ β
[ 2m
n− 1
+ ∆− δ + ∆
n− 1
(
n− 2− (∆− δ)
)]
=
2mθ
n− 1
+ θ (∆− δ) + ∆
n− 1
[
β (n− 1)− θ(∆− δ + 1)
]
,
as β > 0. Moreover, the equality holds in (2.10) if and only if G ∈ Sn or G ∈ Γ.
Next, we assume that β > θ. We consider the following two cases:
Case 1: di = n− 1. In this case
β di + θmi = β (n− 1) + θ
2m− (n− 1)
n− 1
=
2mθ
n− 1
+ β (n− 1)− θ,
and so it is clear that the equality holds in (2.10) as ∆ = n− 1.
Case 2: di ≤ n− 2. Then there is at least one vertex non-adjacent to vi in G. Let G′ be the
graph obtained from the graph G by adding edges between vi and the vertices non-adjacent
to vi in G. Let d′i and m
′
i be the degree of the vertex vi and the average degree of the
vertices adjacent to the vertex vi in G′, respectively. Then d′i = n− 1 and hence G′ ∈ Sn.
Now,
βd′i + θm
′
i = β(n− 1) + θ
(2m+ 2(n− di − 1)− (n− 1)
n− 1
)
= β (n− 1)− θ + 2θ (m+ n− di − 1)
n− 1
. (2.11)
Let pi be the average degree of the vertices non-adjacent to vertex vi in the graph G. Hence
β d′i + θm
′
i − (β di + θmi)
= β (d′i − di) + θ (m′i −mi)
= β (n− di − 1) + θ
(
2m+ (n− di − 1)− (n− 1)
n− 1
−mi
)
= β(n− di − 1) + θ
(
dimi + (n− di − 1)(pi + 1)
n− 1
−mi
)
.
A. Alhevaz et al.: On the Aα-spectral radius of connected graphs 13
Since β ≥ θ > 0 and ∆− δ ≤ n− 2, we have β (n− 1) ≥ θ (∆− δ + 1). Moreover, we
have mi ≤ ∆ and pi ≥ δ for any vertex vi ∈ V (G). Using these results, we obtain
βdi + θmi
= β di − θ +
2θ (m+ n− di − 1)
n− 1
+ θ
(
mi −
dimi + (n− di − 1)(pi + 1)
n− 1
)
=
2mθ
n− 1
+
di
n− 1
(
β (n− 1)− θ
)
+ θ
(
1− di
n− 1
)
(mi − pi)
≤ 2mθ
n− 1
+
di
n− 1
(
β (n− 1)− θ
)
+ θ
(
1− di
n− 1
)
(∆− δ) (2.12)
=
2mθ
n− 1
+ θ (∆− δ) + di
n− 1
(
β (n− 1)− θ − θ (∆− δ)
)
≤ 2mθ
n− 1
+ θ (∆− δ) + ∆
n− 1
(
β (n− 1)− θ (∆− δ + 1)
)
(2.13)
as di ≤ ∆. The first part of the proof is done.
Now, suppose that equality in (2.10) holds with β > θ. Then all the above inequalities
must be equalities. If di = n− 1, then G ∈ Sn. Otherwise, di ≤ n− 2. From the equality
in (2.12), we have mi = ∆ and pi = δ. Since β > θ, we have β (n− 1) > θ (∆− δ + 1).
From the equality in (2.13), we have di = ∆. Therefore all the vertices those are adjacent
to the vertex vi are of degree ∆ and those are non-adjacent to the vertex vi are of degree δ.
Hence G ∈ Γ.
Conversely, let G ∈ Sn. Then ∆ = n− 1 and hence
max
vj∈V
{βdj + θmj} =
2mθ
n− 1
+ β (n− 1)− θ
=
2mθ
n− 1
+ θ (∆− δ) + ∆
n− 1
[
β (n− 1)− θ(∆− δ + 1)
]
.
Let G ∈ Γ. Then 2m = ∆(∆+ 1) + (n−∆− 1) δ and hence
max
vj∈V
{βdj +θmj} = (β+θ)∆ =
2mθ
n− 1
+θ (∆−δ)+ ∆
n− 1
[
β (n−1)−θ(∆−δ+1)
]
.
This completes the proof.
Corollary 2.15. Let G be a graph of order n with m edges and let α ≥ 12 . Let ∆ and δ are
respectively, the maximum degree and the minimum degree of G. Then
max
1≤j≤n
{
αdj + (1− α)mj
}
≤
2m (1− α)
n− 1
+
αn− 1
n− 1
∆ + (1− α)
(
1− ∆
n− 1
)
(∆− δ) (2.14)
with equality if and only if G ∈ Sn or G ∈ Γ.
14 Ars Math. Contemp. 23 (2023) #P1.06
Combining Theorem 2.3 with Corollary 2.15, we get the following result, which gives
an upper bound for the generalized adjacency spectral radius λ(Aα(G)), in terms of the
order n, the size m, the maximum degree ∆, the minimum degree δ and the parameter α.
Theorem 2.16. Let G be a graph of order n, with m edges and let α ≥ 12 . Let ∆ and δ are
respectively, the maximum degree and the minimum degree of G. Then
λ(Aα(G)) ≤
2m (1− α)
n− 1
+
αn− 1
n− 1
∆ + (1− α)
(
1− ∆
n− 1
)
(∆− δ) .
If α ∈
(
1
2 , 1
)
and G is connected, equality holds if and only if G ∼= Kn.
The following result gives a Nordhaus–Gaddum type upper bound for the generalized
adjacency spectral radius λ(Aα(G)), in terms of the order n, the size m, the minimum
degree δ, the maximum degree ∆ and the parameter α.
Theorem 2.17. Let G be a graph of order n, with m edges and let α ≥ 12 . Let ∆ and δ are
respectively, the maximum degree and the minimum degree of G. Then
λ(Aα(G)) + λ(Aα(Ḡ)) ≤ n− 1 +
(1− α)(∆− δ)
n− 1
(
n+ δ −∆− 1 + αn− 1
1− α
)
.(2.15)
If α ∈
(
1
2 , 1
)
and G is connected, equality holds if and only if G = Kn.
Proof. Following Theorem 2.16, we have
λ(Aα(G)) + λ(Aα(Ḡ)) ≤ (1− α)
2m+ 2m̄
n− 1
+
αn− 1
n− 1
(∆ + ∆̄)
+ (1− α)
(
1− ∆
n− 1
)
(∆− δ) + (1− α)
(
1− ∆̄
n− 1
)(
∆̄− δ̄
)
= (1− α)n+ αn− 1
n− 1
(∆− δ + n− 1)
+ (1− α)(∆− δ)
(
1− ∆
n− 1
+
δ
n− 1
)
= n− 1 + (1− α)(∆− δ)
n− 1
(
n+ δ −∆− 1 + αn− 1
1− α
)
,
since m+ m̄ = n(n−1)2 , ∆̄ = n− 1− δ and δ̄ = n− 1−∆.
Now, we consider the equality case in (2.15). If G is regular, then both sides of (2.15)
are equal to n − 1. Now, assume that equality occurs in (2.15) for G. Then the equalities
must hold in (2.15) for both G and Ḡ. Hence G ∼= Kn.
3 A general upper bound for the generalized adjacency spectral ra-
dius
In this section, we obtain a general upper bound for the generalized adjacency spectral
radius in terms of vertex degrees and arbitrary positive real numbers bi. If we replace bi by
some graph parameters, then we can derive some upper bounds for λ(Aα(G)), in terms of
vertex degrees. For this we need the following result:
A. Alhevaz et al.: On the Aα-spectral radius of connected graphs 15
Lemma 3.1 ([7]). Let D = (di,j) be an n×n irreducible non-negative matrix with spectral
radius σ and let Ri(D) =
n∑
j=1
di,j be the i-th row sum of D. Then
min{Ri(D) : 1 ≤ i ≤ n} ≤ σ ≤ max{Ri(D) : 1 ≤ i ≤ n} . (3.1)
Moreover, if the row sums of D are not all equal, then the both inequalities in (3.1) are
strict.
The following result gives an upper bound for λ(Aα(G)), in terms of vertex degrees
and the arbitrary positive real numbers bi.
Theorem 3.2. Let G be a connected graph of order n and 0 < α < 1. Let d1 ≥ d2 ≥
· · · ≥ dn be the vertex degrees of G. Then
λ(Aα(G)) ≤ max
1≤i≤n

αdi +
√
α2d2i +
4
bi
∑
j:j∼i
bj(1− α)(αdj + (1− α)b′j)
2
 , (3.2)
where bi ∈ R+ and b′i = 1bi
∑
j:j∼i
bj . Moreover, the equality holds if and only if αd1 +(1−
α)b′1 = αd2 + (1− α)b′2 = · · · = αdn + (1− α)b′n.
Proof. Let B = diag(b1, b2, . . . , bn), where bi ∈ R+ are positive real number. Since the
matrices Aα(G) and B−1Aα(G)B are similar and similar matrices have same spectrum,
it follows that if λ(Aα(G)) is the largest eigenvalue of Aα(G), then it is also the largest
eigenvalue of B−1Aα(G)B. Let x = (x1, x2, . . . , xn)T be an eigenvector corresponding
to the eigenvalue λ(Aα(G)) of B−1Aα(G)B. We assume that one eigencomponent xi is
equal to 1 and the other eigencomponents are less than or equal to 1. The (i, j)-th entry of
B−1Aα(G)B is 
αdi if i = j,
(1− α) bj
bi
if j ∼ i,
0 otherwise.
We have
B−1Aα(G)Bx = λ(Aα(G))x. (3.3)
From the i-th equation of (3.3), we have
λ(Aα(G))xi = αdixi + (1− α)
∑
j:j∼i
bj
bi
xj ,
i.e., λ(Aα(G)) = αdi + (1− α)
∑
j:j∼i
bj
bi
xj . (3.4)
Again from the j-th equation of (3.3),
λ(Aα(G))xj = αdjxj + (1− α)
∑
k:k∼j
bk
bj
xk.
16 Ars Math. Contemp. 23 (2023) #P1.06
Multiplying both sides of (3.4) by λ(Aα(G)) and substituting this value λ(Aα(G))xj , we
get
λ2(Aα(G)) = αdiλ(Aα(G)) + (1− α)
∑
j:j∼i
bj
bi
αdjxj + (1− α) ∑
k:k∼j
bk
bj
xk

= αdiλ(Aα(G)) + α(1− α)
∑
j:j∼i
bjdj
bi
xj + (1− α)2
∑
j:j∼i
∑
k:k∼j
bk
bi
xk
≤ αdiλ(Aα(G)) + α(1− α)
∑
j:j∼i
bjdj
bi
+ (1− α)2
∑
j:j∼i
bjb
′
j
bi
(3.5)
= αdiλ(Aα(G)) +
∑
j:j∼i
bj(1− α)(αdj + (1− α)b′j)
bi
,
as bi b′i =
∑
j:j∼i bj . Hence we get the upper bound.
Suppose that the equality holds in (3.2). Then all inequalities in the above argument
must be equalities. Since 0 < α < 1, from equality in (3.5), we get xj = 1 for all j such
that j ∼ i, and xk = 1 for all k such that k ∼ j and j ∼ i. From the above, one can easily
prove that xi = 1 for all i ∈ V (G), that is, αd1 + (1− α)b′1 = αd2 + (1− α)b′2 = · · · =
αdn + (1− α)b′n.
Conversely, let G be a connected graph such that αd1+(1−α)b′1 = αd2+(1−α)b′2 =
· · · = αdn + (1 − α)b′n (bi ∈ R+). Since λ(Aα(G)) = λ(B−1Aα(G)B), then by
Lemma 3.1, we obtain
λ(Aα(G)) = αdℓ + (1− α) b′ℓ
= max
1≤i≤n
αdi +
√
α2d2i +
4
bi
∑
j:j∼i bj(1− α)(αdj + (1− α)b′j)
2

for 1 ≤ ℓ ≤ n.
Taking bi = di in (3.2), and noting that b′i =
1
bi
∑
j:j∼i bj =
1
di
∑
j:j∼i dj = mi, we
obtain the following upper bound for λ(Aα(G)), in terms of vertex degrees and average
vertex 2-degrees.
Corollary 3.3. Let G be a connected graph of order n having vertex degrees di, average
vertex 2-degrees mi (1 ≤ i ≤ n) and 0 < α < 1. Then
λ(Aα(G)) ≤ max
1≤i≤n
αdi +
√
α2d2i +
4(1−α)
di
∑
j:j∼i dj [αdj + (1− α)mj ]
2
 .
Equality holds if and only if αd1+(1−α)m1 = αd2+(1−α)m2 = · · · = αdn+(1−α)mn.
Taking bi =
√
di in (3.2), and noting that b′i =
1
bi
∑
j:j∼i bj =
1√
di
∑
j:j∼i
√
dj = m
′
i
(say), we obtain the following upper bound for λ(Aα(G)), in terms of vertex degrees and
m
′
i.
A. Alhevaz et al.: On the Aα-spectral radius of connected graphs 17
Corollary 3.4. Let G be a connected graph of order n having vertex degrees di and let
m
′
i =
1√
di
∑
j:j∼i
√
dj , 1 ≤ i ≤ n and 0 < α < 1. Then
λ(Aα(G)) ≤ max
1≤i≤n
αdi +
√
α2d2i +
4(1−α)√
di
∑
j:j∼i
√
dj [αdj + (1− α)m
′
j ]
2
 .
Equality holds if and only if αd1+(1−α)m
′
1 = αd2+(1−α)m
′
2 = · · · = αdn+(1−α)m
′
n.
Taking bi = 1 in (3.2), and noting that b′i =
1
bi
∑
j:j∼i bj =
∑
j:j∼i 1 = di, we obtain
the following upper bound for λ(Aα(G)), in terms of vertex degrees and average vertex
2-degrees. We note that this upper bound was recently obtained in [15].
Corollary 3.5 ([15]). Let G be a connected graph of order n having vertex degrees di,
average vertex 2-degrees mi (1 ≤ i ≤ n) and 0 < α < 1. Then
λ(Aα(G)) ≤ max
1≤i≤n
{
αdi +
√
α2d2i + 4(1− α)dimi
2
}
.
Equality holds if and only if d1 = d2 = · · · = dn.
Taking bi = mi in (3.2), and noting that b′i =
1
mi
∑
j:j∼i mj = m̄i, we obtain the
following upper bound for λ(Aα(G)), in terms of vertex degrees and the quantity m̄i.
Corollary 3.6. Let G be a connected graph of order n having vertex degrees di, average
vertex 2-degrees mi (1 ≤ i ≤ n) and 0 < α < 1. Then
λ(Aα(G)) ≤ max
1≤i≤n
αdi +
√
α2d2i +
4(1−α)
mi
∑
j:j∼i mj [αdj + (1− α)m̄j ]
2
 ,
where m̄i = 1mi
∑
j:j∼i mj . Equality holds if and only if αd1 + (1 − α)m̄1 = αd2 +
(1− α)m̄2 = · · · = αdn + (1− α)m̄n.
Taking bi = di +mi, bi = di +
√
mi, bi =
√
di +mi, bi =
√
di +
√
mi, bi = 1√di ,
bi =
1√
mi
, bi = 1d2i , bi = d
2
i , etc, and proceeding similarly as above we can obtain some
other new upper bounds for λ(Aα(G)).
4 Relation between ω(G), γ(G) and the generalized adjacency eigen-
values
For a graph G, define ω(G) and γ(G), the clique number and the independence number of
G to be the numbers of vertices of the largest clique and the largest independent set in G,
respectively. In this section, we give bounds for clique number and independence number
of (regular) graph G involving generalized adjacency eigenvalues.
The following lemma, due to Motzkin and Straus [19], links the spectrum of graphs to
its structure.
Lemma 4.1 ([19]). Let F =
{
x = (x1, x2, . . . , xn)
T | xi ≥ 0,
n∑
i=1
xi = 1
}
. Then
1− 1
ω(G)
= max
x∈F
⟨x,Ax⟩.
18 Ars Math. Contemp. 23 (2023) #P1.06
The following result gives a lower bound for ω(G), in terms of the size m, the gen-
eralized adjacency spectral radius λ(Aα(G)), the maximum degree ∆ and the parameter
α.
Theorem 4.2. Let G be a graph of order n, with m edges and maximum degree ∆. Then
ω(G) ≥ 2(1− α)
2m
2(1− α)2m− (λ(Aα(G))− α∆)2
.
Proof. Let x = (x1, x2, . . . , xn)T be the normalized eigenvector corresponding to λ(Aα(G)).
Then
λ(Aα(G)) = α
n∑
i=1
dix
2
i + 2(1− α)
∑
j:j∼i
xixj
≤ α∆
n∑
i=1
x2i + 2(1− α)
∑
j:j∼i
xixj
= α∆+ 2(1− α)
∑
j:j∼i
xixj .
Since λ(Aα(G)) ≥ α(∆ + 1), for α ∈ [0, 12 ], (see [20]), by Cauchy-Schwarz inequality,
we obtain
(λ(Aα(G))− α∆)2 ≤
2(1− α) ∑
j:j∼i
xixj
2 ≤ 2(1− α)2m
2 ∑
j:j∼i
x2ix
2
j
 .
Note that (x21, x
2
2, . . . , x
2
n)
T ≥ 0 and x21 + x22 + · · ·+ x2n = 1. Hence, by Lemma 4.1, we
have
2
∑
j:j∼i
x2ix
2
j ≤ 1−
1
ω(G)
,
then
(λ(Aα(G))− α∆)2
2(1− α)2m
≤ 1− 1
ω(G)
,
that is,
ω(G) ≥ 2(1− α)
2m
2(1− α)2m− (λ(Aα(G))− α∆)2
.
This completes the proof.
Note that Theorem 4.2 extends the Theorem 4.1 proved in [12] for the signless Lapla-
cian spectral radius to generalized adjacency spectral radius.
The following result gives a lower bound for ω(G), when G is a regular graph, in terms
of the order n, the second smallest generalized adjacency eigenvalue λn−1 = λn−1(Aα(G))
and the parameter α
A. Alhevaz et al.: On the Aα-spectral radius of connected graphs 19
Theorem 4.3. Let G be a r-regular graph of order n ≥ 3. Then
ω(G) ≥ (1− α)n
2
(1− α) (n2 − nr) + S2(αr − λn−1)
,
where S = minyi ̸=0
1
|yi| and un−1 = (y1, y2, . . . , yn)
T is the normalized eigenvector
corresponding to λn−1, the second smallest eigenvalue of Aα(G).
Proof. Since G is a r-regular graph, we have λ(Aα(G)) = r and the normalized eigenvec-
tor corresponding to λ(Aα(G)) is u1 = e√n , where e = (1, 1, . . . , 1)
T . Let Θ = Sn
and x = en + Θun−1. Then Θyi ≥ −
1
n (i = 1, 2, . . . , n). Since
∑n
i=1 λi(G) =
2αm = αnr and n ≥ 3, we have λ(Aα(G)) ̸= λn−1(G) and ⟨e, un−1⟩ = 0. So,
x ∈
{
(x1, x2, . . . , xn)
T ;xi ≥ 0,
∑n
i=1 xi = 1
}
. By Lemma 4.1, we have
⟨x, Aαx⟩ = α⟨x, Dx⟩+ (1− α)⟨x, Ax⟩
≤ rα⟨x,x⟩+ (1− α)
(
1− 1
ω(G)
)
= αr
(
1
n
+Θ2
)
+ (1− α)
(
1− 1
ω(G)
)
.
On the other hand
⟨x, Aαx⟩ =
〈 e
n
+Θun−1, Aα
( e
n
+Θun−1
)〉
=
〈 e
n
,Aα
e
n
〉
+
〈 e
n
,AαΘun−1
〉
+
〈
Θun−1, Aα
e
n
〉
+ ⟨Θun−1, AαΘun−1⟩
=
nd
n2
+Θ2λn−1.
Then
d
n
+Θ2λn−1 ≤ αr
(
1
n
+Θ2
)
+ (1− α)
(
1− 1
ω(G)
)
,
that is,
ω(G) ≥ 1− α
(1− α)
(
1− rn
)
+Θ2(αr − λn−1)
.
Since Θ = Sn and S = minyi ̸=0
1
|yi| , we have
ω(G) ≥ (1− α)n
2
(1− α) (n2 − nr) + S2(αr − λn−1)
.
This completes the proof.
Note that Theorem 4.3 extends the Theorem 4.4 proved in [12] for the signless Lapla-
cian spectral radius to generalized adjacency spectral radius.
Consider two sequences of real numbers ξ1 ≥ ξ2 ≥ · · · ≥ ξn and η1 ≥ η2 ≥ · · · ≥ ηt
with t < n. The second sequence is said to interlace the first one whenever
ξi ≥ ηi ≥ ξn−t+i,
20 Ars Math. Contemp. 23 (2023) #P1.06
for i = 1, 2, . . . , t. The interlacing is called tight if there exists an integer k ∈ [0, t] such
that ξi = ηi for 1 ≤ i ≤ k and ξn−t+i = ηi for k + 1 ≤ i ≤ t. Suppose rows and columns
of the matrix M are partitioned according to a partitioning of {1, 2, . . . , n}. The partition
is called regular if each block of M has constant row (and column) sum. The following
lemma can be found in [6].
Lemma 4.4 ([6]). Let B be the matrix whose entries are the average row sums of the blocks
of a symmetric partitioned matrix of M. Then
(i) the eigenvalues of B interlace the eigenvalues of M,
(ii) if the interlacing is tight, then the partition is regular.
Next result gives a lower bound for γ(G), in terms of the order n, the sum of first two
largest generalized adjacency eigenvalues, the maximum degree ∆, the minimum degree δ
and the parameter α.
Theorem 4.5. Let G be a simple graph of order n with at least one edge, with minimum
degree δ and maximum degree ∆. Let λ1(G) and λ2(G) are respectively the first and the
second largest eigenvalue of Aα(G). If λ1(G) + λ2(G)− (1 + α)δ ≤ 0, then
γ(G) ≥ λ1(G) + λ2(G)− (1 + α)δ
δ
× n∆
λ1(G) + λ2(G)− 2∆
. (4.1)
Proof. Let G be a simple graph with order n and a partition V (G) = V1 ∪ V2. Let Gi
(i = 1, 2) be the subgraph of G induced by Vi with ni < n vertices and average degree ri
(n1 + n2 = n). Let ti =
∑
v∈Vi
d(v)
ni
for i = 1, 2. Note that
Aα(G) =
(
A11 A12
A21 A22
)
=
(
αD11 + (1− α)A(G1) (1− α)A12
(1− α)A21 αD22 + (1− α)A(G2)
)
,
where D11 = diag(d(v1), . . . , d(vn1)), D22 = diag(d(vn1+1), . . . , d(vn)) and A21 =
AT12. Put M =
(
mij
ni
)
, where mij is the sum of the entries in Aij(G). Hence
M =
(
αt1 + (1− α)r1 (1− α)(t1 − r1)
(1− α)(t2 − r2) αt2 + (1− α)r2
)
and
|ϕI −M | = ϕ2 − (αt1 + (1− α)r1 + αt2 + (1− α)r2)ϕ
− (1− α)2(t1 − r1)(t2 − r2) + (αt1 + (1− α)r1)(αt2 + (1− α)r2).
Then by Lemma 4.4, we have ϕ1(M) ≤ λ1(G) and ϕ2(M) ≤ λ2(G), hence
ϕ1(M) + ϕ2(M) = αt1 + (1− α)r1 + αt2 + (1− α)r2 ≤ λ1(G) + λ2(G).
Note that 2(n2t2−n1t1) = n2(t2+r2)−n1(t1+r1), and hence n2t2−n1t1 = n2r2−n1r1.
Let VG1 be the largest independent set of G, then r1 = 0 and γ(G) = 0, we have
r2 = t2 − n1n2 t1, and
αt1 + αt2 + (1− α)
(
t2 −
n1
n2
t1
)
= αt1 + t2 − (1− α)
n1
n2
t1 ≤ λ1(G) + λ2(G).
A. Alhevaz et al.: On the Aα-spectral radius of connected graphs 21
By n = n1 + n2, we get
λ1(G) + λ2(G)− t2 − αt1
t1
n ≥ λ1(G) + λ2(G)− t2 − t1
t1
n1.
Since G has at least one edge, n1 < n. Also we have δ ≤ t1, t2 ≤ ∆, hence
λ1(G) + λ2(G)− (1 + α)δ
δ
n ≥ λ1(G) + λ2(G)− 2∆
∆
n1.
Thus
γ(G) = n1 ≥
λ1(G) + λ2(G)− (1 + α)δ
δ
× n∆
λ1(G) + λ2(G)− 2∆
.
This completes the proof.
Again, we note that Theorem 4.5 extends the Theorem 4.5 proved in [12] for the sign-
less Laplacian spectral radius to generalized adjacency spectral radius.
Remark 4.6. Note that if λ1(G) + λ2(G) − (1 + α)δ > 0, then λ1(G)+λ2(G)−(1+α)δδ ×
n∆
λ1(G)+λ2(G)−2∆ < 0, and the inequality in (4.1) is trivial. Hence, we add the restriction
λ1(G) + λ2(G) − (1 + α)δ ≤ 0, in Theorem 4.5. One can easily see that there exists
graphs with the property that λ1(G) + λ2(G) − (1 + α)δ ≤ 0. For example, we have
specAα(Kn) = {n − 1, αn − 1[n−1]}. Hence, λ1(K3) + λ2(K3) − (1 + α)δ(K3) =
2 + 3α− 1− 2(1 + α) = α− 1 ≤ 0.
If G is an r-regular graph, then λ1(G) = r and ∆ = δ = r. Hence, by Theorem 4.5,
we get the following bound.
Corollary 4.7. Let G be a simple r-regular graph of order n with at least one edge. Then
γ(G) ≥ n(λ2(G)− αr)
λ2(G)− r
,
where λ2(G) is the second largest eigenvalue of Aα(G).
5 Some conclusions
As mentioned in the introduction, for α = 0, the generalized adjacency matrix Aα(G) is
same as the adjacency matrix A(G) and for α = 12 , twice the generalized adjacency ma-
trix Aα(G) is same as the signless Laplacian matrix Q(G). Therefore, if in particular, we
put α = 0 and α = 12 , in all the results obtained in Sections 2, 3 and 4, we obtain the
corresponding bounds for the adjacency spectral radius λ(A(G)) and the signless Lapla-
cian spectral radius λ(Q(G)). We note most of these results we obtained in Section 2, 3
and 4 has been already discussed for the adjacency spectral radius λ(A(G)) or/and for the
signless Laplacian spectral radius λ(Q(G)). Therefore, in this setting our results are the
generalization of these known results.
ORCID iDs
Abdollah Alhevaz https://orcid.org/0000-0001-6167-607X
Maryam Baghipur https://orcid.org/0000-0002-9069-9243
Hilal Ahmad Ganie https://orcid.org/0000-0002-2226-7828
Kinkar Chandra Das https://orcid.org/0000-0003-2576-160X
22 Ars Math. Contemp. 23 (2023) #P1.06
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