EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6549
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

The Energy of Cameron-Walker Graphs

Alper Ülker1, Tahsin Oner2, Aiyared Iampan3,∗, Burak Ordin2

1 Department of Mathematics and Computer Science, Istanbul Kültür University,
34156 Istanbul, Turkey
2 Department of Mathematics, Faculty of Science, Ege University, 35100 Izmir, Turkey
3 Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang,
Phayao 56000, Thailand

Abstract. The Cameron-Walker graphs are the graphs for which their matching number equals
their induced matching number. In this paper, we study lower bounds for the graph energy in terms
of induced matching numbers for the general graphs with equal matching and induced matching
number. Moreover, we study the lower bounds of the graph energy of Cameron-Walker graphs.

2020 Mathematics Subject Classifications: 05C50, 05C70, 05C35

Key Words and Phrases: Cameron-Walker graph, matching number, graph energy, lower bound

1. Introduction

The energy of a graph is a measure that directly connects to Hückel Theory and is
defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. Let
G be a simple, undirected graph with vertex set V (G) and edge set E(G). A matching in
G is a subset M ⊆ E(G) such that for any e1, e2 ∈ M , we have e1 ∩ e2 = ∅. The matching
number, denoted by m(G), is the maximum size of a matching in G. A matching M in
a graph G is called an induced matching if, for any distinct e1, e2 ∈ M , there exists no
e ∈ E(G) such that e∩e1 ̸= ∅ and e∩e2 ̸= ∅. The induced matching number of G, denoted
by im(G), is the maximum size of an induced matching in G. The adjacency matrix A(G)
of a graph G = (V (G), E(G)) is a symmetric matrix with entries 0 and 1. The energy of
a graph G with |V (G)| = n is defined as

ε(G) =

n∑
i=1

|λi|

where λ1, λ2, . . . , λn are the eigenvalues of the adjacency matrix of G.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6549

Email addresses: a.ulker@iku.edu.tr (A. Ülker), tahsin.oner@ege.edu.tr (T. Oner),
aiyared.ia@up.ac.th (A. Iampan), burak.ordin@ege.edu.tr (B. Ordin)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



A. Ülker et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6549 2 of 8

The matching polynomial of a graph G is given by

µ(G) = µ(G,λ) =
∑
k≥0

(−1)km(G, k)λn−2k

where m(G, k) is the number of k-matchings in G. The matching energy (ME) of G is
defined as the sum of the absolute values of the zeros of its matching polynomial. This
topic has been extensively studied [1–5].

A fundamental question in spectral graph theory concerns lower bounds on the energy
of a graph in terms of its matching number. In [6], Wong et al. proved that

ε(G) ≥ 2im(G)

for any graph G. Moreover, they established the refined bound

ε(G) ≥ 2im(G) +

√
5

5
c1(G),

where c1(G) denotes the number of disjoint odd cycles in G. Later, in [7], Ashraf improved
this result by showing that

ε(G) ≥ 2im(G) + c0(G),

where c0(G) represents the number of disjoint odd cycles of length at least 5.
In this paper, we extend these results to the class of graphs known as Cameron-Walker

graphs in which the matching number equals the induced matching number. In Section
3, for a given constant induced matching number, we determine the graph with minimum
energy using matching energy. In Section 4, we establish new lower bounds on the energy
of graphs where the matching and induced matching numbers are equal.

2. Preliminaries

This chapter is devoted to the definitions and previous results that will be used in the
rest of the paper.

Definition 1. Let G be a simple undirected graph of order n. If mk(G) is the number of
k-matchings of G, k ∈ {0, 1, 2, ...,

⌊
n
2

⌋
}, then the matching energy of G is

ME(G) =
2

π

∞∫
0

1

x2
ln[

∑
k≥0

mk(G)x2k]dx.

By the monotonicity of the logarithm, one can define a quasi-order relation “⪰” as the
following: Let G1 and G2 be two graphs. Then G1 ⪰ G2 ⇐⇒ mk(G1) ≥ mk(G2) for all k.

The following theorem can provide the matching energy of a graph.



A. Ülker et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6549 3 of 8

Theorem 1. ([1], Theorem 1) Let G be a simple graph and let σ1, σ2, ..., σn be the zeros
of the matching polynomial of G. Then,

ME(G) =
n∑

i=1

|σi|.

If e = uv is an edge in G, then the equality m(G; k) = m(G−e, k)+m(G−u−v; k−1)
holds. So m(G; k) increases whenever an edge is added to the graph. Hence, we can give
the following result.

Theorem 2. ([1], Theorem 3) Let G be a graph and e be its edge. If G− e is obtained by
removing e from G and keeping all the vertices of G remaining, then ME(G−e) < ME(G).

The following theorem states that energy and matching energy coincide for trees.

Theorem 3. ([1], Theorem 2) If a graph G has no cycle, then its matching energy and
energy are equal.

In [8], Arizmendi et al. studied the vertex energy in a graph energy. The following
theorem gives the lower bound of a vertex energy in terms of its degree.

Theorem 4. ([8], Theorem 3.3) Let G be a connected graph with at least one edge. Then
for all vi ∈ V (G)

εG(vi) ≥
di
∆
.

Equality holds if and only if G is isomorphic to the complete bipartite graph Kd,d.

3. Graph parameters and energy

In this section, we study the graphs with minimal energy with induced matching
number im(G). Moreover, we provide lower bounds on graph energy with respect to the
induced matching number.

Lemma 1. Let G be any graph and H be its induced subgraph. Then ε(H) ≤ ε(G).

Proof. Since H is an induced subgraph of G, then adjacency matrix of G is

A(G) =

[
A(H) X
XT A(G\H)

]
which implies that A(H) is the principal submatrix of A(G). Thus, the eigenvalues of
A(H) are less than those of A(G). Therefore, ε(H) < ε(G). If G = H, then A(G\H) = 0,
thus eigenvalues of A(H) and A(G) are same and ε(H) = ε(G).

Next, we define a new graph called rake-graph RGn over n vertices, and we show that
this graph has the minimum energy among the graphs with induced matching number n.



A. Ülker et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6549 4 of 8

Definition 2. Let n ≥ 2 be an integer. A graph RGn, called a rake graph, is a graph on
2n + 1 vertices constructed as follows: take n disjoint copies of the complete graph K2,
and connect each of their vertices to a single additional vertex. Formally,

V (RGn) = {v0} ∪
n⋃

i=1

{ui, vi},

E(RGn) =

n⋃
i=1

{(ui, vi), (v0, vi)}.

Equivalently, the rake graph can be viewed as n disjoint edges (i.e., K2 components), each
of which is attached to a single central vertex v0 via one of its endpoints.

It is easy to verify that both the matching number and the induced matching number
of RGn are equal to n.

Example 1. Figure 1 depicts the graph RG4. The matching number and the induced
matching number of this graph are 4. This graph has minimal energy among the connected
graphs with induced matching number 4 (See Theorem 5).

Figure 1: Graph RG4.

If im(G) = 1, then it is clear that K2 is the graph with minimum energy among the
graphs with im(G) = 1. In the following theorem, we study the graphs with minimum
energy among those with a constant induced matching number im(G) > 2.

Theorem 5. Let G be a connected graph with induced matching number im(G) > 2. Then
the energy and the matching energy of G are minimized by the graph RGim(G).

Proof. In the view of Theorem 2, the graph G must be a tree since it has a mini-
mal matching energy. By Theorem 3, it is enough to show that this graph has minimal
matching energy with the given induced matching number. Let M be the set of induced
matchings. Then G[M ] forms a disjoint union of graphs K2 with the number of such



A. Ülker et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6549 5 of 8

graphs equal to im(G). Since M is a set of induced matchings, this implies that for each
ei ∈ M , there exists at least one edge e′i adjacent to ei with e′i /∈ M . Moreover, e′i is
adjacent to only ei, otherwise e′i would be a common edge of two edges in M and a con-
tradiction. By minimality, we can consider one of the vertices of the edges ei = uivi of M
is a pendant vertex, say ui.

Since m(G; k) = m(G − ei; k) +m(G − ui − vi; k − 1) for all ei = uivi ∈ M , then the
subgraph after removing edges ei ∈ M , we have a cut-vertex left for minimality. Since the
matching energy of a tree coincides with its energy, the graph RGim(G) has the minimum
energy among the graphs with induced matching number im(G) ≥ 2.

Theorem 6. Let G = (V,E) be a graph with induced matching number im(G). Then the
followings hold:

(1) For any graph G, ε(G) ≥ 2im(G), and equality holds if and only if G is isomorphic
to the disjoint union of graphs K2.

(2) If G is connected and im(G) ≥ 2, then ε(G) ≥ ε(RGim(G)).

Proof. (1) Assume that M ⊆ E be an induced matching set. The induced subgraph
on G[M ] forms a disjoint union of graphs K2. Since each K2 contributes 2 to the ε(G),
then by Lemma 1, we get ε(G) ≥ 2im(G). If we assume G is isomorphic to the disjoint
union of graphs K2, i.e., G ∼= nK2, this implies that im(G) = n. Since ε(G) = 2n, then
equality holds.

(2) Assume that G is a connected graph. Since the minimal energy graph with induced
matching number im(G) ≥ 2 is RGim(G) by Theorem 5, then it is clear that ε(G) ≥
ε(RGim(G)) for any connected graph G.

4. Energy of graphs with equal matching and induced matching number

Cameron and Walker gave a characterization of undirected connected finite simple
graphs that satisfy m(G) = im(G) in [9]. These graphs are known as Cameron-Walker
graphs and have previously been studied from a commutative algebra perspective, partic-
ularly in relation to edge ideals and their algebraic invariants [10]. The authors gave a
slightly modified definition of these graphs. Next, we provide the definitions of this graph
in the light of [10].

Definition 3. A finite simple and connected graph G satisfies m(G) = im(G) if G is one
of the following:

(i) G is a star graph,

(ii) G is a star triangle,

(iii) G is a finite graph consisting of a connected supporting bipartite graph with vertex
partition U ⊔V such that there is at least one leaf edge attached to each vertex u ∈ U
and that there may be some pendant triangles connected to each vertex v ∈ V .



A. Ülker et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6549 6 of 8

Definition 4. A finite connected simple graph G is called a Cameron–Walker graph if
im(G) = m(G) and if G is neither a star nor a star triangle.

The following proposition provides a lower bound for graphs that include an induced
subgraph with im(G) = m(G). These graphs do not belong to the class of Cameron-Walker
graphs.

Proposition 1. Let G be a graph containing an induced subgraph consisting of n pendant
triangles sharing a common vertex. Then the energy of G satisfies ε(G) ≥ 2n+ 1.

Proof. Let G′ be the induced subgraph of G consisting of n pendant triangles sharing a

common vertex. The adjacency matrix A(G′) of G′ is

[
0 1
1T Q

]
. Since rank(Q) = 2n and

1 is 1×2n matrix with all 1’s, it is clear that rank(A(G′)) = 2n+1. Hence ε(G′) ≥ 2n+1
by Lemma 2 in [11]. Thus, by Lemma 1, we get ε(G) ≥ 2n+ 1.

In the next proposition, we give a lower bound for the energy of a Cameron-Walker
graph in terms of its matching number and maximum degree.

Proposition 2. The graph RGim(G) is the minimal energy Cameron-Walker graph with
(induced) matching number im(G) ≥ 2.

Proof. The graph RGim(G) is a Cameron-Walker graph with supporting bipartite graph
K1,im(G). And the number of leaves is im(G), which implies (induced) matching number
is im(G). Thus, by Theorem 5, RGim(G) is a Cameron-Walker graph of minimal energy.

Proposition 3. Let G be a Cameron-Walker graph. Then the followings hold:
(1) If G is a C3-free graph with m disjoint leaf edges, then ε(G) ≥ ε(RGm).
(2) If G has m disjoint leaf edges and n disjoint pendant triangles, then ε(G) ≥ 2m+4n.
(3) If G has m disjoint leaf edges and n pendant triangles, then ε(G) ≥ 2m+ 2n+ 1.

Proof. (1) A Cameron-Walker graph is a connected graph. Since G is C3-free and
possessesm leaves, then there are no pendant triangles and the (induced) matching number
of G is m. Hence, by Proposition 2, we conclude that ε(G) ≥ ε(RGm).

(2) Let L be the set of disjoint leaf edges and let T be the set of disjoint pendant
triangles. In a Cameron-Walker graph, it is clear that L∩T = ∅. Since each leaf edge has
energy equal to 2 and each pendant triangle has energy equal to 4, by Lemma 1 it follows
that ε(G) ≥ 2m+ 4n.

(3) Now, without loss of generality, we assume that all the pendant triangles of G have
a common vertex. Therefore, the induced subgraph on n pendant triangles has energy at
least 2n + 1 by Proposition 1. Since each leaf edge contributes 2 to the energy, then we
get that ε(G) ≥ 2m+ 2n+ 1.

Example 2. The following graph G is a Cameron-Walker graph with 4 leaf edges and 2
pendant triangles. The energy ε(G) ∼= 19.55 and (induced) matching number is 6.

In the following theorem, we give a lower bound for a Cameron-Walker graph in terms
of its matching number and number of triangles.



A. Ülker et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6549 7 of 8

Figure 2: A Cameron-Walker Graph with 2 pendant triangles

Theorem 7. If G is a Cameron-Walker graph with maximum degree ∆, then

ε(G) ≥ 2m(G) +
3

∆
C3(G),

where m(G) is the matching number and C3(G) is the number of disjoint pendant triangles
in G.

Proof. Let M be the maximum matching set of G. And let C = {C1, C2, ..., Cr} be
the set of disjoint triangles of G. Since G is a Cameron-Walker graph, the set M consists
of disjoint pendant edges and edges of triangles with end vertices that have degree 2.
Without loss of generality, let us assume that all the pendant triangles are disjoint in
G. Let the sets V (G) and V (G[M ]) be the vertex sets of G and the induced graph on
M , respectively. Thus, ε(G) ≥ ε(G[M ]) + ε(V (G)\G[M ]) by Lemma 1. It is clear that
ε(G[M ]) = 2m(G). And from Theorem 4, for any vi ∈ V (G)\V (G[M ]) we have ε(vi) ≥ 3

∆
since the degree of a vertex in V (G)\V (G[M ]) has degree at least 3. Thus, it follows that
ε(G) ≥ 2m(G) + 3

∆C3(G).

5. Conclusion

We investigated the graph energy of a special class of graphs—those for which the
matching number equals the induced matching number, with particular focus on Cameron-
Walker graphs. Using spectral graph-theoretic techniques, we established new lower
bounds for the energy in terms of the induced matching number and provided compar-
isons with known bounds in the literature. Our results demonstrate that the structural
constraints of Cameron-Walker graphs yield meaningful spectral restrictions, thereby in-
fluencing their energy in predictable ways. The Cameron-Walker graphs can have unique
energy properties due to their controlled structure. Further studies can explore their other
spectral properties.

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments
and insightful feedback, which helped improve the quality and clarity of this work. This



A. Ülker et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6549 8 of 8

research was supported by University of Phayao and Thailand Science Research and In-
novation Fund (Fundamental Fund 2025, Grant No. 5027/2567).

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