EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 3, 2023, 1731-1746
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

New Improved Bounds for Signless Laplacian Spectral
Radius and Nordhaus-Gaddum Type Inequalities for

Agave Class of Graphs

Malathy V1, Kalyani Desikan1,∗

1 Division of Mathematics, School of Advanced Sciences, Vellore Institute of Technology,
Chennai, India

Abstract. Core-satellite graphs Θ(c, s, η) ∼= Kc ▽ (ηKs) are graphs consisting of a central clique
Kc (the core) and η copies of Ks (the satellites) meeting in a common clique. They belong to the
class of graphs of diameter two. Agave graphs Θ(2, 1, η) ∼= K2 ▽ (ηK1) belong to the general class
of complete split graphs, where the graphs consist of a central clique K2 and η copies of K1 which
are connected to all the nodes of the clique. They are the subclass of Core-satellite graphs. Let
µ(G) be the spectral radius of the signless Laplacian matrix Q(G). In this paper, we have obtained
the greatest lower bound and the least upper bound of signless Laplacian spectral radius of Agave
graphs. These bounds have been expressed in terms of graph invariants like m the number of
edges, n the number of vertices, δ the minimum degree, ∆ the maximum degree, and η copies
of the satellite. We have made use of the approximation technique to derive these bounds. This
unique approach can be utilized to determine the bounds for the signless Laplacian spectral radius
of any general family of graphs. We have also obtained Nordhaus-Gaddum type inequality using
the derived bounds.

2020 Mathematics Subject Classifications: 05C50

Key Words and Phrases: Spectral radius, signless Laplacian spectral radius, Core-satellite
graphs, Agave graphs, Complete split graph

1. Introduction

Hierarchical products of graphs, developed through iterative hierarchical products of
complete graphs, are used as models for real-world networks. Core-satellite graphs can be
redesigned to resemble some important characteristics of a complex network to exhibit a
hierarchical structure [14, 18]. Estrada [6] demonstrated how certain classes of real-world
networks can be modeled using core-satellite graphs. This family of graphs has topologi-
cal features of both small world and scale-free networks. Using heuristic analysis, Xu and
Zhang [20] investigated the effects of cover time. They showed that the network topology

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4806

Email addresses: malathy.viswanathan2015@vit.ac.in (Malathy V), kalyanidesikan@vit.ac.in (K. Desikan)

https://www.ejpam.com 1731 © 2023 EJPAM All rights reserved.



Malathy V, Kalyani Desikan / Eur. J. Pure Appl. Math, 16 (3) (2023), 1731-1746 1732

discussed by them in their work, having both small world and scale-free features, exhibits
the most minimal cover time.
Agave graphs are formed by connecting η disjoint vertices to both vertices of a complete
graph K2. They belong to the general class of complete split graphs which are formed by
the join of central clique Kr and a set of (n− r) independent vertices [6, 7]. Nair et al. [1]
distinctly picturised the complete split graphs which belong to the family of generalized
core-satellite graphs. The generalized core-satellite graphs belong to the larger family of
graphs of diameter two.
The significant features of Agave graphs are identified from the article by Ernesto and
Eusebio [6]. The distance-sum heterogeneity index φ(G), plays a key role in the structural
analysis of complex networks. This allows an interpretation of the Wiener and Balaban
index. It is found to be useful in the analysis of molecular graphs. Furthermore, it is found
that there is proximity with a node’s closeness centrality and average path length in the
analysis of complex networks. This index highlights more of the structural properties of
a network which helps us to understand the functional and dynamic processes in complex
systems. The authors have conjectured that among graphs with a specific number of nodes
φ(G) is the maximum for graphs with structures resembling the Agave graph.
Due to its enhanced effectiveness, simplicity, and better performance, the signless Lapla-
cian spectral radius is utilized to examine graph properties. When compared to spectra
of other graph matrices, the spectrum of the signless Laplacian matrix is thoroughly in-
vestigated and studied extensively [4, 11, 19].
Through different methods, the bounds are obtained for µ(Q(G)). Y. Chen and L.Wang
[3] provided two sharp upper bounds for µ(Q(G)) in terms of the maximum degree and
the minimum degree of the graph G and employed a new technique to obtain another
sharp upper bound. In [5], Xing and Zhou provided a generalized theorem for the upper
bound and lower bound for the spectral radius of a non-negative matrix using its row
sums. They have applied these bounds to various non-negative matrices associated with
graphs which includes the adjacency matrix, the signless Laplacian matrix, the distance
matrix, the distance signless Laplacian matrix, and the reciprocal distance matrix. In
[21], Zhang et al. determined the largest signless Laplacian spectral radius among the bi-
cyclic graphs with perfect matchings. In [10], the authors determined sharp upper bounds
for the spectral radius and signless Laplacian spectral radius of a uniform hypergraph in
terms of the average 2 degrees or degrees of vertices, degree and the minimum degree of
the vertices of G. Moreover, they proposed a new proving technique to obtain a sharp
upper bound for µ(Q(G)). In [13], Feng and Yu studied the family of connected graphs
with prescribed order and diameter and then determined the extremal graph that has the
maximal signless Laplacian spectral radius.
Y. Hong and Shu [8] obtained a sharp upper bound of the Nordhaus-Gaddum type for the
chromatic numbers of G and Gc, respectively. Huiqing Liu et al. [11] obtained a sharp up-
per bound for the Nordhaus-Gaddam type relation of the Laplacian spectral radius L(G)
by making use of the relation µ(A(G)) ≤ µ(Q(G)). Shuchao and Tian [9] and Shi [19],
have derived sharp bounds for Nordhaus-Gaddam type of relation in terms n, m, δ(G),
and ∆(G). Nikiforov [15–17], conjectured and obtained improved bounds on Nordhaus-



Malathy V, Kalyani Desikan / Eur. J. Pure Appl. Math, 16 (3) (2023), 1731-1746 1733

Figure 1: Agave graph K2 ▽ 4K1 and K2 ▽ ηK1

Gaddum type inequality. In [2], Mustapha Aouchiche and Pierre Hansen mentioned many
results on lower and upper bounds in their survey article on Nordhaus-Gaddum type in-
equality on the sum and the product of many other graph invariants of a graph and its
complements.
Motivated by the above-mentioned results, we have implemented some unique techniques
to obtain tight upper and lower bounds for the signless Laplacian spectral radius for the
Agave class of graphs. We have utilized these bounds to obtain the corresponding tight
bounds for Nordhaus-Gaddam type inequality.

2. Preliminaries

Here we mention some preliminaries, definitions, and results which will be used to
derive our main results.

Let G = (V,E) be a simple, connected graph with n vertices and m edges. Let δ(G) = δ
and ∆(G) = ∆ be the minimum and the maximum degree of vertices of G, respec-
tively. Let Gc be the complement of graph G. Let A(G) be the adjacency matrix
of the graph G and the degree diagonal matrix D(G) = diag (d(v1), d(v2), ..., d(vn)) is
the diagonal matrix of vertex degrees. For any n × n real symmetric matrix M , by
Geršgorin’s Theorem, its eigenvalues are non-negative real numbers. Let ρ(G) be the
spectral radius of adjacency matrix A(G), with non-increasing sequence of real eigenvalues
ρ(G) = ρ1(G) ≥ ρ2(G) ≥ ... ≥ ρn(G). Let µ(G) be the signless Laplacian spectral radius
of the matrix Q(G) = D(G) + A(G), which is irreducible, symmetric, and non-negative.
Thus the matrix Q(G) is positive and semi-definite. It has non-increasing sequence of real
eigenvalues µ(Q(G)) = µ1(Q(G)) ≥ µ2(Q(G)) ≥ ... ≥ µn(Q(G)) = 0.

Definition 1. [6] The join (or complete product) G1 ▽ G2 of graphs G1 and G2 is the
graph obtained from G1 ∪G2 by joining every vertex of G1 with every vertex of G2.

Definition 2. [6] Core-satellite graphs Θ(c, s, η) ∼= Kc ▽ (ηKs) are the graphs consisting
of η copies of Ks (the satellites) meeting in a common clique Kc (the core), where c ≥ 1,
s ≥ 1 and η ≥ 2.

Core-satellite graphs have central nodes in a network that are connected among them-
selves. Also, there are a few cliques of the same size which are connected to the central



Malathy V, Kalyani Desikan / Eur. J. Pure Appl. Math, 16 (3) (2023), 1731-1746 1734

core but not connected among themselves. For v ∈ V , the degree of v, written as d(v),
is the number of edges incident on v. The vertices in the satellite group have minimum
degree δ and the vertices of the central clique (core) have the maximum degree ∆. Agave
graphs are graphs with Kc as K2, Ks as K1 and η is the number of copies of Ks. This
class belongs to the general class of complete split graphs [6].

The following results are useful in proving the main results.

Lemma 1. [19] Let B be a real symmetric n × n matrix and µ be an eigenvalue of B
with an eigenvector x all of whose entries are non-negative. Denote the ith row sum of B
by Ri(B). Then

min
1≤i≤n

Ri(B) ≤ µ(B) ≤ max
1≤i≤n

Ri(B). (1)

Moreover, if all entries of x are positive then either of equalities holds if and only if
row sums of B are all equal.

Lemma 2. [19] Let M be a real symmetric n × n matrix and λ be an eigenvalue of M
with an eigenvector x whose entries are all non-negative. Let P be any polynomial. Then

min
1≤i≤n

Si(P (M)) ≤ P (λ) ≤ max
1≤i≤n

Si(P (M))

where Si(P (M)) is the row sum of the ith row of the matrix P (M). Moreover, if all entries
of x are positive then either of the equalities holds if and only if the row sums of P (M)
are all equal.

Lemma 3. [9] Let G = (V,E) be a simple graph. Then

√
2 min
v∈V (G)

√
d2 (v) +

∑
uv∈E(G)

d (u) ≤ µ(G) ≤
√
2 max
v∈V (G)

√
d2 (v) +

∑
uv∈E(G)

d (u)

(2)

Moreover, ifG is connected, both the equalities hold if and only if 2
(
d2 (v) +

∑
uv∈E(G) d (u)

)
is the same for all v ∈ V (G).

Lemma 4. [12] The signless Laplacian spectrum of Ka ∨Kb is{
(3b+ a− 2)±

√
(b+ a− 2)2 + 4ab

2
, ba−1, (a− b− 2)b−1

}

where an exponent indicates the multiplicity of the corresponding signless Laplacian eigen-
value.



Malathy V, Kalyani Desikan / Eur. J. Pure Appl. Math, 16 (3) (2023), 1731-1746 1735

3. Main Results

Theorem 1. Let GA be an Agave graph with n vertices, m edges, and η copies of the
satellite graph. Let ∆ and δ be the maximum degree and minimum degree of GA. Also(

2δ2 + 2(2m− ηδ)
)1/2 ≤ µ (GA) ≤

(
2∆2 + 2(2m−∆)

)1/2
(3)

Moreover, both the equalities hold if and only if n = 3 and η = 1.

Proof. Using Lemma 3, the lower bound is attained when the vertex v is in the satellite
graph.

√
2 min
v∈V (G)

√
d2(v) +

∑
uv∈E(GA)

d(u)

=
√
2

√√√√√δ2 +

2m− d(u)−
∑

uv/∈E(GA)

d(u)



=
√

2δ2 + 2 (2m− δ − (η − 1)δ)

=
√

2δ2 + 2(2m− ηδ) ≤ µ(GA).

Similarly, the upper bound is attained when v is the vertex in the core graph. Also for
vertices in the core graph, that is v ∈ C, we have d(v) = ∆ = (n− 1) and∑

uv/∈E(G)

d (u) = 0

since v ∈ C is connected to all the vertices of the graph GA. Therefore

√
2 max
u∈V (G)

√
d2(v) +

∑
uv∈E(GA)

d(u) =
√
2

√
∆2 + (2m− d (v)−

∑
uv/∈E(G)

d (u))

=
√

2∆2 + 2(2m−∆) ≥ µ (GA)

Hence, √
2δ2 + 2(2m− ηδ) ≤ µ(GA) ≤

√
2∆2 + 2(2m−∆)

Moreover, both the equalities hold for n = 3 and η = 1, that is when δ = ∆.



Malathy V, Kalyani Desikan / Eur. J. Pure Appl. Math, 16 (3) (2023), 1731-1746 1736

Example 1. According to the result derived in Theorem 1, for n = 7, m = 11, η = 5,
∆ = 6, δ = 2 and µ(GA) = 8.5311, we have

5.6568 ≤ µ(GA) ≤ 10.198 (4)

Remark 1. The following relationships between the parameters in the graph GA hold:

(i) m = (n+ η − 1) = (2n− 3) (ii) ∆ = (n− 1) (iii) (δ +∆) = (n+ 1)

(iv) (∆− δ) = (η − 1) = (n− 3) (v) (n− η) = δ = 2.

We prove the following Lemmas, which are used to prove the theorems.

Lemma 5. For the graph GA,

µ(GA) >
(
2δ2 + 2(2m− ηδ) + (n+ 1)η

)1/2
>
(
2δ2 + 2(2m− ηδ) + nη

)1/2
for all n > 6.

Proof. To prove this we make use of Lemma 4. We consider the signless Laplacian
spectral radius for Ka ∨Kb

µ (GA) =
(3b+ a− 2) +

√
(b+ a− 2)2 + 4ab

2
(5)

Substituting b = 2 and a = η, where η = n − 2 and making use of Remark 1, we reduce
equation (5) in terms of n as

µ (GA) =

{
(n+ 2) +

√
(n2 + 4n− 12)

2

}
Squaring the above expression we have

(µ (GA))
2 =

{
(n+ 2)2 + (n2 + 4n− 12) + 2(n+ 2)

√
(n2 + 4n− 12)

4

}
In the above expression, using the following binomial approximation√

(n2 + 4n− 12) ≃
(
1 +

n

2

(
4

n
− 12

n2

))
when

1

2

∣∣∣∣ 4n − 12

n2

∣∣∣∣ < 1

for n ≥ 3, we get

(µ (GA))
2 ≃

{
(n+ 2)2 + (n2 + 4n− 12) + 2(n+ 2)

(
1 + n

2

(
4
n − 12

n2

))
4

}



Malathy V, Kalyani Desikan / Eur. J. Pure Appl. Math, 16 (3) (2023), 1731-1746 1737

=

(
n2 + 4n− 6

n
− 3

)
.

To prove

µ(GA) >
(
2δ2 + 2(2m− ηδ) + (n+ 1)η

)1/2
consider

(µ(GA))
2 −

((
2δ2 + 2(2m− ηδ) + (n+ 1)η

)1/2)2
Expressing the second term in the above expression in terms of n and using the approxi-
mation

(µ(GA))
2 ≃

(
n2 + 4n− 6

n
− 3

)
.

we obtain

(µ (GA))
2 −

(
n2 + 3n+ 2

)
≃
(
n− 6

n
− 5

)
> 0

for n > 6. Hence proved.

Lemma 6. For the graph GA,

µ (GA) >
(
2∆2 + 2(2m−∆)− nη

)1/2
for n ≥ 3.

Proof. We follow the procedure adopted to prove Lemma 5. The above inequality is
proved by reducing the RHS expression in terms of n and on using the approximation for
(µ (GA))

2, we obtain

(µ (GA))
2 −

(
n2 + 4n− 8

)
≃
(
5− 6

n

)
> 0

for n ≥ 3.

Lemma 7. For the graph GA,

µ (GA) <
(
2δ2 + 2(2m− ηδ) + (n+ 2)η

)1/2
for n ≥ 3.

Proof. The proof is on the same lines as that of Lemma 5 and we have

(µ (GA))
2 −

(
n2 + 4n

)
≃
(
− 6

n
− 3

)
< 0

for n ≥ 3.



Malathy V, Kalyani Desikan / Eur. J. Pure Appl. Math, 16 (3) (2023), 1731-1746 1738

Corollary 1. Using Lemma 5, we obtain the approximation for µ (GA) as

µ (GA) ≃
(
n2 + 4n− 4.2

)1/2
(6)

for all n > 6. We now consider

µ(GA) ≃
(
2δ2 + 2(2m− ηδ) + (n+ 1)η + γ

)1/2
Substituting δ = 2, m = (2n− 3) and η = (n− 2) in the above equation, we get

µ(GA) ≃ (8 + 2(2(2n− 3)− 2(n− 2)) + (n+ 1)(n− 2) + γ)1/2

µ(GA) ≃
(
n2 + 3n+ 2 + γ

)1/2
For n = 7, we have

µ(GA) = 8.5311 = (72 + γ)1/2

For n = 8,
µ(GA) = 9.5826 = (90 + γ)1/2

For n = 9,
µ(GA) = 10.6235 = (110 + γ)1/2

For n = 10,
µ(GA) = 11.6569 = (132 + γ)1/2

For n = 11,
µ(GA) = 12.6847 = (156 + γ)1/2

By observing the pattern, we find that γ ≃ (n− 6.2) for all n > 6.

µ(GA) ≃
(
n2 + 3n+ 2 + (n− 6.2)

)1/2
=
(
n2 + 4n− 4.2

)1/2
for n > 6.

In the following Theorem, we determine the values of κ and κ′ in order to obtain the
greatest lower bound and the least upper bound for µ (GA).

Theorem 2. For the Agave graph GA,

µ (GA) >
(
2δ2 + 2(2m− ηδ) + (n+ 2)η − κ

)1/2
(7)

where 5 ≤ κ ≤ (n2 − 4) and the greatest lower bound is attained when κ = 5 also

µ (GA) <
(
2∆2 + 2(2m−∆)− nη + κ′

)1/2
(8)

where 4 ≤ κ′ ≤ n(n− 2) and the least upper bound is attained when κ′ = 4.



Malathy V, Kalyani Desikan / Eur. J. Pure Appl. Math, 16 (3) (2023), 1731-1746 1739

Proof. Consider Lemma 7,

µ(GA) <
(
2δ2 + 2(2m− ηδ) + (n+ 2)η

)1/2
Reducing this expression in terms of n, we have

µ(GA) ≃
(
n2 + 4n− 4.2

)1/2
<
(
n2 + 4n

)1/2
where this is true for all n > 6. We determine a positive number κ such that

µ(GA) ≃
(
n2 + 4n− 4.2

)1/2
>
(
2δ2 + 2(2m− ηδ) + (n+ 2)η − κ

)1/2
From the above inequality, we have(

n2 + 4n− 4.2
)1/2

>
(
n2 + 4n− κ

)1/2
This implies that

κ > 4.2 ⇒ κ = 5

We can easily verify that the expression
(
2δ2 + 2(2m− ηδ) + (n+ 2)η − 5

)1/2
is the great-

est lower bound. Consider

µ(GA) ≃
(
2δ2 + 2(2m− ηδ) + (n+ 2)η − 5 + 1

)1/2
to be the greatest lower bound of µ(GA). That is

µ(GA) ≃
(
n2 + 4n− 4.2

)1/2
<
(
n2 + 4n− 4

)1/2
.

This is a contradiction for all n > 6.

Hence
(
2δ2 + 2(2m− ηδ) + (n+ 2)η − 5

)1/2
is the greatest lower bound. We observe that

the greatest lower bound occurs when κ = 5 and the loose lower bound is attained when
κ = (n2 − 4).

Similarly, in order to determine the least upper bound for µ(GA), we consider the up-
per bound given in equation (3) of Theorem 1. Consider Lemma 6,

µ(GA) >
(
2∆2 + 2(2m−∆)− nη

)1/2
Reducing the above expression in terms of n, we have

µ(GA) ≃
(
n2 + 4n− 4.2

)1/2
>
(
n2 + 4n− 8

)1/2
In order to determine the least upper bound, we need to find the value κ′ such that

µ(GA) <
(
2∆2 + 2(2m−∆)− nη + κ′

)1/2
=
(
n2 + 4n− 8 + κ′

)1/2



Malathy V, Kalyani Desikan / Eur. J. Pure Appl. Math, 16 (3) (2023), 1731-1746 1740

µ(GA) ≃
(
n2 + 4n− 4.2

)1/2
<
(
n2 + 4n− 8 + κ′

)1/2
This implies that

κ′ > 3.8

⇒ κ′ = 4

Therefore
µ(GA) <

(
2∆2 + 2(2m−∆)− nη + 4

)1/2
It can easily be shown that the above expression is the least upper bound.

Consider
µ(GA) <

(
2∆2 + 2(2m−∆)− nη + 3

)1/2
=
(
n2 + 4n− 5

)1/2
to be the least upper bound. From this, we notice that

µ(GA) ≃
(
n2 + 4n− 4.2

)1/2
<
(
n2 + 4n− 5

)1/2
This is a contradiction. Therefore

µ(GA) <
(
2∆2 + 2(2m−∆)− nη + 4

)1/2
. (9)

is the least upper bound.

Clearly, we observe that the least upper bound occurs when κ′ = 4 and the loose up-
per bound is attained when κ′ = n(n− 2).

We prove that the lower bound and the upper bound are the greatest lower bound and
the least upper bound for µ(GA) by making use of the relation between the parameters
given in Remark 1.

Lemma 8. For the graph GA, the lower bound and upper bound are improved as

(2δ2 + 2(2m− ηδ) + (n+ 2)η − 5)1/2 < µ (GA) (10)

and (
2∆2 + 2(2m−∆)− (nη − 4)

)1/2
> µ (GA) (11)

We verify that when ((n + 2)η − 5) is added to the expression in the lower bound in
equation (3) of Theorem 1, we get the greatest lower bound and when the term (nη − 4)
is subtracted from the expression in the upper bound in equation (3) of Theorem 1, we
obtain the least upper bound for the signless Laplacian spectral radius µ (GA).

Proof. Let us assume

L.B∗ =
(
2δ2 + 2(2m− ηδ) + (ϵ1 + 1)

)1/2
= (2δ2 + 2(2m− ηδ) + ((n+ 2)η − 5 + 1))1/2



Malathy V, Kalyani Desikan / Eur. J. Pure Appl. Math, 16 (3) (2023), 1731-1746 1741

to be the greatest lower bound of µ (GA) and let

U.B = (2∆2 + 2(2m−∆)− (nη − 4))1/2.

By making use of the relationship between the parameters given in Remark 1, we observe
that (L.B∗)2 − (U.B)2 =

(2δ2 + 2(2m− ηδ) + (n+ 2)η − 4)− (2∆2 + 4m− 2∆− (nη − 4))

= 2(δ +∆)(δ −∆)− 2ηδ + nη + 2η − 4 + 2(η + 1) + nη − 4

= 2(n+ 1)(1− η)− 4η + nη + 2η + 2η − 4 + 2 + nη − 4

= 2(n− η)− 2nη + 4 + 2nη − 8 = 0

This implies L.B∗ = U.B which is a contradiction.

Therefore the following expression is the greatest lower bound

(2δ2 + 2(2m− ηδ) + (n+ 2)η − 5)1/2 < µ (GA) .

Similarly, we prove that ϵ2 = (nη − 4) is the maximum integral value required to obtain
the least upper bound.

Let us assume that

U.B∗ =
(
(2∆2 + 2(2m−∆)− (ϵ2 + 1)

)1/2
=
(
(2∆2 + 2(2m−∆)− (nη − 3)

)1/2
is the least upper bound and let L.B = (2δ2 + 2(2m− ηδ) + (n+ 2)η − 5)1/2.

Consider (U.B∗)2 − (L.B)2 =

(2∆2 + 4m− 2∆− (nη − 3))− (2δ2 + 2(2m− ηδ) + (n+ 2)η − 5)

= 2(δ +∆)(δ −∆) + 2ηδ − 2∆− (nη − 3) + (n+ 2)η − 5 = 0

Using the relationship between the parameters given in Remark 1, we observe that
U.B∗ − L.B = 0 which implies U.B∗ = L.B, which is a contradiction. Therefore, we
conclude that U.B is the greatest upper bound. Therefore we have

L.B =
(
2δ2 + 2(2m− ηδ) + (n+ 2)η − 5

)1/2
< µ (GA) <(

2∆2 + 2(2m−∆)− (nη − 4)
)1/2

= U.B. (12)

Remark 2. By using the improved bounds derived in Theorem 2, for the graph K2▽5K1,
with n = 7, m = 11, η = 5, ∆ = 6, δ = 2 and µ(GA) = 8.5311, we have

8.4852 < µ(GA) < 8.544



Malathy V, Kalyani Desikan / Eur. J. Pure Appl. Math, 16 (3) (2023), 1731-1746 1742

4. Bounds of the Nordhaus-Gaddam type

In this section, we obtain the upper and the lower bounds on µ (GA)+µ (Gc
A) (Nordhaus-

Gaddam type inequality) for the signless Laplacian spectral radius of the Agave graphs in
terms of the maximum degree ∆, the minimum degree δ, the order n, and the size m of
the graph G.

Theorem 3. Let GA be an Agave graph with n vertices, m edges and η copies of the
satellite graph K1. Let ∆ and δ be the maximum degree and minimum degree of GA,
respectively.

2

(
(4n2 − 20n+ 40) + ((n+ 2)η − κ)

2

)1/2

< µ (GA) + µ (Gc
A) < 2

(
(6n2 − 22n+ 28)− (nη − κ′)

2

)1/2

(13)

for 5 ≤ κ ≤ (n2 − 4) and 4 ≤ κ′ ≤ (n(n− 2)).

Proof. Consider the family of Agave graphs GA. Let η be the number of satellites K1

join with K2 and let Gc
A be the complement having mc edges, where

mc = nC2 −m, ∆c = δc = (n− 3)

From equation (7) of Theorem 2, we consider the lower bound

µ(GA) >
(
2δ2 + 2(2m− ηδ) + (n+ 2)η − κ

)1/2
and the signless Laplacian spectral radius of complement of GA

µ(Gc
A) = µ(Kη−1) = (2(∆− δ)2 + 2(n(n− 1)− 2m− (n− 3))1/2

We have
µ(GA) + µ(Gc

A) >
(
2δ2 + 2(2m− ηδ) + (n+ 2)η − κ

)1/2
+
(
2(∆− δ)2 + 2(n(n− 1)− 2m− (n− 3)

)1/2
= g(m)

(14)

Then
dg

dm
< 0

if and only if{
2√

(2δ2 + 4m− 2ηδ + (n+ 2)η − κ)

}
>

{
(−2)√

(2(∆− δ)2 + 2n(n− 1)− 4m− 2(n− 3))

}



Malathy V, Kalyani Desikan / Eur. J. Pure Appl. Math, 16 (3) (2023), 1731-1746 1743

By reducing ∆ and δ in terms of n and solving for m, we have

m >

(
(4n2 − 12n+ 8)− ((n+ 2)η − κ)

8

)
Substituting in (14), we get

µ (GA) + µ (Gc
A) >(

2δ2 + 4

((
4n2 − 12n+ 8)− ((n+ 2)η − κ

)
8

)
− 2ηδ) + (n+ 2)η − κ

)1/2

+

(
2(∆− δ)2 + 2n(n− 1)− 4

((
4n2 − 12n+ 8)− ((n+ 2)η − κ

)
8

)
− 2(n− 3)

)1/2

Therefore

µ (GA) + µ (Gc
A) > 2

(
(4n2 − 20n+ 40) + ((n+ 2)η − κ)

2

)1/2

for 5 ≤ κ ≤ (n2 − 4).
(15)

Similarly, we derive the condition for the existence of the upper bound. We consider the
upper bound given in equation (8) of Theorem 2

µ (GA) <
(
2∆2 + 2(2m−∆)− (nη − κ′)

)1/2
(16)

µ (Gc
A) =

(
2(∆− δ)2 + 2(2mc −∆c)

)1/2
where mc = nC2 −m

µ (Gc
A) =

(
2(∆− δ)2 + 2(n(n− 1)− 2m− (n− 3)

)1/2
(17)

µ (GA) + µ (Gc
A) <

(
2∆2 + 4m− 2∆− (nη − κ′)

)1/2
+
(
2(∆− δ)2 + 2n(n− 1)− 4m− 2(n− 3)

)1/2
= g(m) (18)

Then
dg

dm
> 0

if and only if{
2√

(2∆2 + 4m− 2∆− (nη − κ′))

}
<

{
(−2)√

(2(∆− δ)2 + 2n(n− 1)− 4m− 2(n− 3))

}



Malathy V, Kalyani Desikan / Eur. J. Pure Appl. Math, 16 (3) (2023), 1731-1746 1744

Reducing ∆ and δ in terms of n and solving for m, we get

m <

(
(2n2 − 10n+ 20) + (nη − κ′)

8

)
Substituting the value of m in (18), we get

µ (GA) + µ (Gc
A) < 2

(
(6n2 − 22n+ 28)− (nη − κ′)

2

)1/2

for 4 ≤ κ′ ≤ (n(n− 2)).

Therefore

2

(
(4n2 − 20n+ 40) + ((n+ 2)η − κ)

2

)1/2

< µ (GA) + µ (Gc
A) < 2

(
(6n2 − 22n+ 28)− (nη − κ′)

2

)1/2

for 5 ≤ κ ≤ (n2 − 4) and 4 ≤ κ′ ≤ n(n− 2).

Hence proved.

Remark 3. It can be easily verified from equation (13) of Theorem 3, for the graph
K2 ▽ 5K1, where n = 7, m = 11, ∆ = 6, δ = 2, η = 5, κ = 5 and κ′ = 4, the bounds are

16.4924 < µ (GA) + µ (Gc
A) < 16.553

5. Discussion

In this paper, we have utilized a new technique to improve the bounds and obtain the
tight upper and lower bounds for the signless Laplacian spectral radius of Agave class of
graphs. These bounds are in terms of the maximum degree ∆, the minimum degree δ,
number of satellites η, n the number of vertices and m the number of edges of the graph
GA. They are exceptionally proximate to µ (GA). Also, we have derived new improved
upper and lower bounds for the Nordhaus-Gaddum type inequality. We anticipate that
the technique used to obtain these bounds will be helpful in determining the tight lower
and the upper bounds for the signless Laplacian spectral radius of any general class of
graphs.

Acknowledgements

The authors are grateful to the reviewers for their valuable comments which have
enhanced the article.



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