# Laplacian Spectral Characterization of Some Unicyclic Graphs

Authors: Lijun Yu; Hui Wang; Jiang Zhou
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

LetW
(
n
;
q
,
m
1
,
m
2
) be the unicyclic graph with n vertices obtained by attaching two paths of lengths m
1 and m
2 at two adjacent vertices of cycle C
q. Let U
(
n
;
q
,
m
1
,
m
2
,
…
,
m
s
) be the unicyclic graph with n vertices obtained by attaching s paths of lengths m
1
,
m
2
,
…
,
m
s at the same vertex of cycle C
q. In this paper, we prove that W
(
n
;
q
,
m
1
,
m
2
) and U
(
n
;
q
,
m
1
,
m
2
,
…
,
m
s
) are determined by their Laplacian spectra when q is even.


---

## Body

## 1. Introduction

LetG be a simple, undirected graph with n vertices. Let A be the adjacency matrix of G and let D be the diagonal matrix of vertex degrees of G. The matrices L
=
D
-
A and Q
=
D
+
A are called theLaplacian matrix andsignless Laplacian matrix of G, respectively. The multiset of eigenvalues of A and L are called theA-spectrum andL-spectrum of G, respectively. The eigenvalues of A and L are called theA-eigenvalues andL-eigenvalues of G, respectively. We use λ
1
(
G
)
⩾
λ
2
(
G
)
⩾
⋯
⩾
λ
n
(
G
) and μ
1
(
G
)
⩾
μ
2
(
G
)
⩾
⋯
⩾
μ
n
(
G
)
=
0 to denote the A-eigenvalues and the L-eigenvalues of G, respectively. Two graphs are said to beL-cospectral (A-cospectral) if they have the same L-spectrum (A-spectrum). A graph G is said to bedetermined by its L-spectrum (A-spectrum) if there is no other nonisomorphic graph L-cospectral (A-cospectral) with G. Let ϕ
A
(
G
,
x
), ϕ
L
(
G
,
x
), and ϕ
Q
(
G
,
x
) denote the characteristic polynomials of the adjacency matrix, the Laplacian matrix, and the signless Laplacian matrix of G, respectively. As usual, P
n, C
n, and K
n stand for the path, the cycle, and the complete graph with n vertices, respectively. Let l
(
G
) denote the line graph of G. A tree is calledstarlike if it has exactly one vertex of degree larger than 2. Let T
a
,
b
,
c denote the starlike tree with a vertex v of degree 3 such that T
a
,
b
,
c
-
v
=
P
a
∪
P
b
∪
P
c.For a connected graphG with n vertices, G is called aunicyclic graph if G has n edges. Which graphs are determined by their spectrum is a difficult problem in the theory of graph spectra. Here, we introduce some results on spectral characterizations of unicyclic graphs. Let U
(
n
;
q
,
m
1
,
m
2
,
…
,
m
s
) be the unicyclic graph with n vertices obtained by attaching s paths of lengths m
1
,
m
2
,
…
,
m
s
(
m
i
⩾
1
) at the same vertex of cycle C
q (see Figure 1). Haemers et al. [1] proved that U
(
n
;
q
,
m
1
) is determined by its A-spectrum when q is odd, and all U
(
n
;
q
,
m
1
) are determined by their L-spectra. It is also known that U
(
n
;
q
,
m
1
) is determined by its A-spectrum when q is even [2]. Liu et al. [3] proved that U
(
n
;
q
,
m
1
,
m
2
) is determined by its L-spectrum. It is known that U
(
n
;
q
,
1,1
,
…
,
1
) is determined by its L-spectrum, and U
(
n
;
q
,
1,1
,
…
,
1
) is determined by its A-spectrum if q is odd (see [4]). Boulet [5] proved that the sun graph is determined by its L-spectrum. Shen and Hou [6] gave a class of unicyclic graphs with even girth that are determined by their L-spectra.Figure 1
Two classes of unicyclic graphs.LetW
(
n
;
q
,
m
1
,
m
2
) be the unicyclic graph with n vertices obtained by attaching two paths of lengths m
1 and m
2
(
m
1
,
m
2
⩾
1
) at two adjacent vertices of cycle C
q (see Figure 1). In this paper, we prove that W
(
n
;
q
,
m
1
,
m
2
) and U
(
n
;
q
,
m
1
,
m
2
,
…
,
m
s
) are determined by their L-spectra when q is even.


## 2. Preliminaries

In this section, we give some lemmas which play important roles throughout this paper.Lemma 1 (see [7]).
LetG be a graph. For the adjacency matrix and the Laplacian matrix, the following can be obtained from the spectrum:(i)
the number of vertices,
(ii)
the number of edges.
For the adjacency matrix, the following follows from the spectrum:(iii)
the number of closed walks of any length.
For the Laplacian matrix, the following follows from the spectrum:(iv)
the number of components,
(v)
the number of spanning trees.Lemma 2 (see [8]).
For a bipartite graphG, one has ϕ
L
(
G
,
x
)
=
ϕ
Q
(
G
,
x
).Lemma 3 (see [8]).
LetG be a graph with n vertices and m edges. Then
(1)
ϕ
A
(
l
(
G
)
,
x
)
=
(
x
+
2
)
m
-
n
ϕ
Q
(
G
,
x
+
2
)
.For a graphG with n vertices, let ϕ
L
(
G
,
x
)
=
l
0
x
n
+
l
1
x
n
-
1
+
⋯
+
l
n. Oliveira et al. determined the first four coefficients of ϕ
L
(
G
,
x
) as follows.Lemma 4 (see [9]).
LetG be a graph with n vertices and m edges, and let d
1
,
d
2
,
…
,
d
n be the degree sequence of G. Then
(2)
l
0
=
1
,
l
1
=
-
2
m
=
-
∑
i
=
1
n
‍
d
i
,
l
2
=
2
m
2
-
m
-
1
2
∑
i
=
1
n
‍
d
i
2
,
l
3
=
1
3
[
-
4
m
3
+
6
m
2
+
3
m
2
∑
i
=
1
n
‍
d
i
2
-
∑
i
=
1
n
‍
d
i
3
-
3
∑
i
=
1
n
‍
d
i
2
+
6
N
G
(
C
3
)
]
,
where N
G
(
C
3
) is the number of triangles in G.For a graphG, thesubdivision graph of G, denoted by S
(
G
), is the graph obtained from G by inserting a new vertex in each edge of G.Lemma 5 (see [8]).
LetG be a graph with n vertices and m edges. Then
(3)
ϕ
A
(
S
(
G
)
,
x
)
=
x
m
-
n
ϕ
Q
(
G
,
x
2
)
.Lemma 6 (see [8]).
Letu be a vertex of G, let N
(
u
) be the set of all vertices adjacent to u, and let C
(
u
) be the set of all cycles containing u. Then
(4)
ϕ
A
(
G
,
x
)
=
x
ϕ
A
(
G
-
u
,
x
)
-
∑
v
∈
N
(
u
)
‍
ϕ
A
(
G
-
u
-
v
,
x
)
-
2
∑
Z
∈
C
(
u
)
‍
ϕ
A
(
G
-
V
(
Z
)
,
x
)
,
where V
(
Z
) is the vertex set of Z.Lemma 7 (see [10]).
Considerϕ
A
(
P
n
,
2
)
=
n
+
1.Lemma 8 (see [1]).
LetG be a graph with n vertices and let v be a vertex of G. Then λ
1
(
G
)
⩾
λ
1
(
G
-
v
)
⩾
λ
2
(
G
)
⩾
λ
2
(
G
-
v
)
⩾
⋯
⩾
λ
n
-
1
(
G
-
v
)
⩾
λ
n
(
G
).Lemma 9 (see [5]).
LetG be a graph with edge set E
(
G
). Then
(5)
μ
1
(
G
)
⩽
max
⁡
{
d
(
u
)
+
d
(
v
)
:
u
v
∈
E
(
G
)
}
,
where d
(
u
) stands for the degree of vertex u.Lemma 10 (see [11]).
For a connected graphG with at least two vertices, one has μ
1
(
G
)
⩾
Δ
(
G
)
+
1, where Δ
(
G
) denotes the maximum vertex degree of G; equality holds if and only if Δ
(
G
)
=
n
-
1.Lemma 11 (see [12]).
LetG be a connected graph with n
⩾
3 vertices and let d
2 be the second maximum degree of G. Then d
2
⩽
μ
2
(
G
).Lemma 12 (see [8]).
LetG be a graph with n vertices and let e be an edge of G. Then μ
1
(
G
)
⩾
μ
1
(
G
-
e
)
⩾
μ
2
(
G
)
⩾
μ
2
(
G
-
e
)
⩾
⋯
⩾
μ
n
-
1
(
G
-
e
)
⩾
μ
n
(
G
)
=
μ
n
(
G
-
e
)
=
0.For a graphG, let N
G
(
M
) denote the number of subgraphs of G which are isomorphic to graph M.Lemma 13 (see [13]).
LetG be a graph and let N
G
(
k
) be the number of closed walks of length k in G. Then
(6)
N
G
(
3
)
=
6
N
G
(
C
3
)
,
N
G
(
5
)
=
30
N
G
(
C
3
)
+
10
N
G
(
C
5
)
+
10
N
G
(
U
(
4
;
3,1
)
)
.


## 3. Main Results

Lemma 14.
LetG be a unicyclic graph with n vertices, and G contains an even cycle C
q. Let H be a graph L-cospectral with G. Then the following statements hold.(1)
H is a unicyclic graph with n vertices, and the girth of H is q.
(2)
The line graphsl
(
G
) and l
(
H
) are A-cospectral.
(3)
The subdivision graphsS
(
G
) and S
(
H
) are A-cospectral, and μ
i
(
G
)
=
λ
i
(
S
(
G
)
) (i
=
1,2
,
…
,
n).Proof.
By Lemma1, H is a unicyclic graph with n vertices, and the girth of H is q. Since q is even, G and H are bipartite. By Lemma 2, one has ϕ
Q
(
G
,
x
)
=
ϕ
L
(
G
,
x
)
=
ϕ
L
(
H
,
x
)
=
ϕ
Q
(
H
,
x
). Lemma 3 implies that line graphs l
(
G
) and l
(
H
) are A-cospectral. By Lemma 5, subdivision graphs S
(
G
) and S
(
H
) are A-cospectral, and μ
i
(
G
)
=
λ
i
(
S
(
G
)
) (i
=
1,2
,
…
,
n).Theorem 15.
The unicyclic graphG
=
W
(
n
;
q
,
m
1
,
m
2
) is determined by its L-spectrum when q is even.Proof.
LetH be any graph L-cospectral with G. By Lemma 14, we know that H is a unicyclic graph with n vertices, the girth of H is q, and l
(
G
) and l
(
H
) are A-cospectral. By Lemmas 1 and 13, we have N
l
(
H
)
(
C
3
)
=
N
l
(
G
)
(
C
3
)
=
2. So the maximum degree of H does not exceed 3. Suppose that there are a
i vertices of degree i
(
i
=
1,2
,
3
) in H. From Lemma 4, we have
(7)
∑
i
=
1
3
‍
a
i
=
n
,
∑
i
=
1
3
‍
i
a
i
=
2
n
,
∑
i
=
1
3
‍
i
2
a
i
=
2
×
3
2
+
4
(
n
-
4
)
+
2
=
4
n
+
4
.
Solving the above equations, we get a
1
=
2
,
a
2
=
n
-
4
,
a
3
=
2. So H and G have the same degree sequence. Then, one of the following holds.(1)
H is the unicyclic graph obtained by attaching two paths of lengths l
1 and l
2 at two nonadjacent vertices of cycle C
q.
(2)
H
=
W
(
n
;
q
,
l
1
,
l
2
); that is, H is the unicyclic graph obtained by attaching two paths of lengths l
1 and l
2 at two adjacent vertices of cycle C
q.
(3)
H is the graph shown in Figure 2.
Next, we discuss each of these three cases listed above.
Case  1 (H is the unicyclic graph obtained by attaching two paths of lengths l
1 and l
2 at two nonadjacent vertices of cycle C
q). Since l
(
G
) and l
(
H
) are A-cospectral, by Lemma 1, l
(
G
) and l
(
H
) have the same number of closed walks of any length. It is not difficult to see that N
l
(
G
)
(
C
5
)
=
N
l
(
H
)
(
C
5
). By Lemma 13, we have N
l
(
H
)
(
U
(
4
;
3,1
)
)
=
N
l
(
G
)
(
U
(
4
;
3,1
)
). Note that m
1
+
m
2
+
q
=
l
1
+
l
2
+
q
=
n. If m
1
⩾
2 or m
2
⩾
2, then N
l
(
G
)
(
U
(
4
;
3,1
)
)
⩾
7 and N
l
(
H
)
(
U
(
4
;
3,1
)
)
⩽
6. If m
1
=
m
2
=
1, then N
l
(
G
)
(
U
(
4
;
3,1
)
)
=
6 and N
l
(
H
)
(
U
(
4
;
3,1
)
)
=
4. Hence N
l
(
H
)
(
U
(
4
;
3,1
)
)
≠
N
l
(
G
)
(
U
(
4
;
3,1
)
), a contradiction.
Case  2 (H is the unicyclic graph W
(
n
;
q
,
l
1
,
l
2
)). From Lemma 14, we know that the subdivision graphs S
(
G
) and S
(
H
) (shown in Figure 3) are A-cospectral. Let p
f
=
ϕ
A
(
P
f
,
x
); from Lemmas 6 and 7, we have
(8)
ϕ
A
(
S
(
G
)
,
x
)
=
x
p
2
m
1
+
2
m
2
+
2
q
-
1
-
(
p
2
m
1
p
2
q
-
2
+
2
m
2
+
p
2
m
2
p
2
q
-
2
+
2
m
1
)
-
2
p
2
m
1
p
2
m
2
,
ϕ
A
(
S
(
G
)
,
2
)
=
2
(
2
m
1
+
2
m
2
+
2
q
)
-
(
2
m
1
+
1
)
(
2
q
+
2
m
2
-
1
)
-
(
2
m
2
+
1
)
(
2
q
+
2
m
1
-
1
)
-
2
(
2
m
1
+
1
)
(
2
m
2
+
1
)
=
-
4
(
m
1
q
+
m
2
q
+
4
m
1
m
2
)
,
ϕ
A
(
S
(
H
)
,
x
)
=
x
p
2
l
1
+
2
l
2
+
2
q
-
1
-
(
p
2
l
1
p
2
q
-
2
+
2
l
2
+
p
2
l
2
p
2
q
-
2
+
2
l
1
)
-
2
p
2
l
1
p
2
l
2
,
ϕ
A
(
S
(
H
)
,
2
)
=
2
(
2
l
1
+
2
l
2
+
2
q
)
-
(
2
l
1
+
1
)
(
2
q
+
2
l
2
-
1
)
-
(
2
l
2
+
1
)
(
2
q
+
2
l
1
-
1
)
-
2
(
2
l
1
+
1
)
(
2
l
2
+
1
)
=
-
4
(
l
1
q
+
l
2
q
+
4
l
1
l
2
)
.
Byϕ
A
(
S
(
G
)
,
2
)
=
ϕ
A
(
S
(
H
)
,
2
), we get -
4
(
m
1
q
+
m
2
q
+
4
m
1
m
2
)
=
-
4
(
l
1
q
+
l
2
q
+
4
l
1
l
2
). By m
1
+
m
2
+
q
=
l
1
+
l
2
+
q
=
n, we get m
1
m
2
=
l
1
l
2. Hence, m
1
=
l
1
,
m
2
=
l
2 or m
1
=
l
2
,
m
2
=
l
1, G and H are isomorphic.
Case  3 (H is the graph shown in Figure 2). It is well known that the largest L-eigenvalue of a path is less than 4, and the largest L-eigenvalue of an even cycle is 4. Lemma 12 implies that μ
2
(
G
)
<
4. Let u and v be the two vertices of degree 3 in H (see Figure 2). If u and v are nonadjacent, there exists an edge e of H such that H
-
e
=
C
q
∪
T
l
1
,
l
2
,
n
-
l
1
-
l
2
-
q
-
1. By Lemmas 10 and 2.12, we get μ
2
(
H
)
⩾
4, a contradiction to μ
2
(
G
)
<
4. So u and v are adjacent.
From Lemma14, we know that the subdivision graphs S
(
G
) and S
(
H
) (shown in Figure 4) are A-cospectral. Let p
f
=
ϕ
A
(
P
f
,
x
); from Lemmas 6 and 7, we have
(9)
ϕ
A
(
S
(
G
)
,
x
)
=
x
p
2
m
1
+
2
m
2
+
2
q
-
1
-
(
p
2
m
1
p
2
q
-
2
+
2
m
2
+
p
2
m
2
p
2
q
-
2
+
2
m
1
)
-
2
p
2
m
1
p
2
m
2
,
ϕ
A
(
S
(
G
)
,
2
)
=
2
(
2
m
1
+
2
m
2
+
2
q
)
-
(
2
m
1
+
1
)
(
2
q
+
2
m
2
-
1
)
-
(
2
m
2
+
1
)
(
2
q
+
2
m
1
-
1
)
-
2
(
2
m
1
+
1
)
(
2
m
2
+
1
)
=
-
4
(
m
1
q
+
m
2
q
+
4
m
1
m
2
)
,
ϕ
A
(
S
(
H
)
,
x
)
=
x
p
2
q
-
1
ϕ
A
(
T
1,2
l
1
,
2
l
2
,
x
)
-
(
p
2
q
-
1
p
2
l
1
+
2
l
2
+
1
+
2
p
2
q
-
2
ϕ
A
(
T
1,2
l
1
,
2
l
2
,
x
)
)
-
2
ϕ
A
(
T
1,2
l
1
,
2
l
2
,
x
)
,
ϕ
A
(
S
(
H
)
,
2
)
=
2
×
2
q
ϕ
A
(
T
1,2
l
1
,
2
l
2
,
2
)
-
[
2
q
(
2
l
1
+
2
l
2
+
2
)
+
2
(
2
q
-
1
)
ϕ
A
(
T
1,2
l
1
,
2
l
2
,
2
)
]
-
2
ϕ
A
(
T
1,2
l
1
,
2
l
2
,
2
)
=
-
4
q
(
l
1
+
l
2
+
1
)
.
Sinceϕ
A
(
S
(
G
)
,
2
)
=
ϕ
A
(
S
(
H
)
,
2
), we have -
4
q
(
l
1
+
l
2
+
1
)
=
-
4
(
m
1
q
+
m
2
q
+
4
m
1
m
2
). By l
1
+
l
2
+
1
=
m
1
+
m
2, we get m
1
m
2
=
0, a contradiction to m
1
,
m
2
>
0.Figure 2
GraphH.Figure 3
Two subdivision graphs.Figure 4
Two subdivision graphs.Here, we describe a classic method to count the number of closed walks of a given length in a graph (see [2, 13, 14]). For a graph G, N
G
(
k
) stands for the number of closed walks of length k in G and N
G
(
M
) stands for the number of subgraphs of G which are isomorphic to graph M. Let ω
k
(
M
) be the number of closed walks of length k of graph M which contains all edges of M, and M
k
(
G
) denotes the set of all connected subgraphs M of G such that ω
k
(
M
)
≠
0. Then
(10)
N
G
(
k
)
=
∑
M
∈
M
k
(
G
)
‍
N
G
(
M
)
ω
k
(
M
)
.Lemma 16.
LetG
=
U
(
n
;
q
,
m
1
,
m
2
,
…
,
m
s
) and G
′
=
U
(
n
;
q
,
l
1
,
l
2
,
…
,
l
s
) be L-cospectral graphs. If q is even, then G and G
′ are isomorphic.Proof.
Ifq is even, by Lemma 14, l
(
G
) and l
(
G
′
) are A-cospectral. From Lemma 1, we get N
l
(
G
)
(
k
)
=
N
l
(
G
′
)
(
k
) for any positive integer k. Suppose m
1
⩽
m
2
⩽
⋯
⩽
m
s, l
1
⩽
l
2
⩽
⋯
⩽
l
s. Let r
i
=
min
⁡
{
m
i
,
l
i
}
(
i
=
1,2
,
…
,
s
). If m
1
≠
l
1, by m
1
+
m
2
+
⋯
+
m
s
=
l
1
+
l
2
+
⋯
+
l
s, we know that M
2
r
1
+
3
(
l
(
G
)
)
=
M
2
r
1
+
3
(
l
(
G
′
)
). For any M
∈
M
2
r
1
+
3
(
l
(
G
)
) and M
≠
U
(
3
+
r
1
;
3
,
r
1
), we have N
l
(
G
)
(
M
)
=
N
l
(
G
′
)
(
M
). Since N
l
(
G
)
(
U
(
3
+
r
1
;
3
,
r
1
)
)
≠
N
l
(
G
′
)
(
U
(
3
+
r
1
;
3
,
r
1
)
), by (10), we get N
l
(
G
)
(
2
r
1
+
3
)
≠
N
l
(
G
′
)
(
2
r
1
+
3
), a contradiction. So we have m
1
=
l
1. Similar to the above arguments, by counting the number of closed walks of length 2
r
i
+
3
(
i
=
2,3
,
…
,
s
), we can get m
i
=
l
i
(
i
=
2,3
,
…
,
s
). Hence G and G
′ are isomorphic.Theorem 17.
The unicyclic graphG
=
U
(
n
;
q
,
m
1
,
m
2
,
…
,
m
s
) is determined by its L-spectrum when q is even.Proof.
LetG
′ be any graph L-cospectral with G. By Lemma 14, G
′ is a unicyclic graph with n vertices, and the girth of G
′ is q. Let v be the vertex of degree s
+
2 in the subdivision graph S
(
G
)
=
U
(
2
n
;
2
q
,
2
m
1
,
2
m
2
,
…
,
2
m
s
); then S
(
G
)
-
v
=
P
2
q
-
1
∪
P
2
m
1
∪
P
2
m
2
∪
⋯
∪
P
2
m
s. Since the largest A-eigenvalue of a path is less than 2, by Lemmas 8 and 14, we get μ
2
(
G
)
=
λ
2
(
S
(
G
)
)
<
2
,
μ
2
(
G
)
<
4. Suppose d
1
⩾
d
2
⩾
⋯
⩾
d
n is the degree sequence of G
′. By Lemma 11, we have d
2
⩽
3. From Lemmas 9 and 10, we get s
+
3
<
μ
1
(
G
)
⩽
s
+
4, d
1
+
d
2
⩾
μ
1
(
G
)
>
s
+
3, and d
1
+
1
<
μ
1
(
G
)
⩽
s
+
4. By d
2
⩽
3, we have s
<
d
1
<
s
+
3.
Ifd
1
=
s
+
2, applying Lemma 4, we have
(11)
∑
i
=
2
n
‍
d
i
=
2
+
2
+
⋯
+
2
︸
n
-
s
-
1
+
1
+
1
+
⋯
+
1
︸
s
,
∑
i
=
2
n
‍
d
i
2
=
2
2
+
2
2
+
⋯
+
2
2
︸
n
-
s
-
1
+
1
2
+
1
2
+
⋯
+
1
2
︸
s
.
Since ∑
i
=
2
n
‍
d
i
2 is minimal if and only if |
d
i
-
d
j
|
≤
1 for any i
,
j
∈
{
2,3
,
…
,
n
}, the degree sequences of G and G
′ are both s
+
2
,
2,2
,
…
,
2
︸
n
-
s
-
1
,
1,1
,
…
,
1
︸
s. Lemma 16 implies that G and G
′ are isomorphic.
Ifd
1
=
s
+
1, by d
1
+
d
2
>
s
+
3 and d
2
<
4, we get d
2
=
3. Suppose that there are a
3 three, a
2 two, and a
1 one in d
2
,
d
3
,
…
,
d
n. By Lemma 4, we have
(12)
∑
i
=
1
3
‍
a
i
+
1
=
n
,
∑
i
=
1
3
‍
i
a
i
+
(
s
+
1
)
=
s
+
2
(
n
-
s
-
1
)
+
(
s
+
2
)
,
∑
i
=
1
3
‍
i
2
a
i
+
(
s
+
1
)
2
=
s
+
4
(
n
-
s
-
1
)
+
(
s
+
2
)
2
.
Solving the above equations, we get a
1
=
2
s
-
1
,
a
2
=
n
-
3
s
,
a
3
=
s. From Lemma 4, we have
(13)
∑
i
=
1
3
‍
i
3
a
i
+
(
s
+
1
)
3
=
s
+
8
(
n
-
s
-
1
)
+
(
s
+
2
)
3
.
s
=
0 or s
=
1 is the solution of the above equation. Then d
1
=
1 or d
1
=
2, a contradiction to d
2
=
3.Thejoin of two graphs G and H, denoted by G
×
H, is the graph obtained from G
∪
H by joining each vertex of G to each vertex of H. Some results on spectral characterizations of graphs obtained by join operation can be found in [15–20]. For a unicyclic graph G, if G is determined by its L-spectrum and G
≠
C
6, then G
×
K
r is determined by its L-spectrum (cf. [18, Theorem 4.4]). Hence, we can obtain the following two results from Theorems 15 and 17.Corollary 18.
LetG
=
W
(
n
;
q
,
m
1
,
m
2
). Then G
×
K
r is determined by its L-spectrum when q is even.Corollary 19.
LetG
=
U
(
n
;
q
,
m
1
,
m
2
,
…
,
m
s
). Then G
×
K
r is determined by its L-spectrum when q is even.



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