On Laplacian Equienergetic Signed Graphs

A signed graph is an ordered pair Γ � (G, σ), where G � (V, E) is a simple unsigned graph called the underlying graph of Γ, and σ: E⟶ +, − { }, called a signing (or a signature), is a sign function from the edge set E to the set +, − { } of signs. *e sign of a signed graph is defined as the product of signs of its edges. A signed graph is said to be positive (respectively, negative) if its sign is positive (respectively, negative), i.e., it contains an even (respectively, an odd) number of negative edges. A signed graph is said to be allpositive (respectively, all-negative) if all its edges are positive (respectively, negative), denoted by (G, +) (respectively(G, − )). A signed graph is said to be balanced if each of its cycles is positive, otherwise unbalanced. Let A(G) � (aij)n×n be adjacency matrix of unsigned graph G, where aij � 1 whenever vertices i and j are adjacent and aij � 0 otherwise. And, let D(G) be its diagonal matrix of vertices degrees; then, L(G) � D(G) − A(G) and Q(G) � D(G) + A(G) are called Laplacian and signless Laplacian matrix of G, respectively. Similarly, for a signed graph Γ � (G, σ), its adjacency matrix A(Γ) � (aij)n×n is defined as


Introduction
A signed graph is an ordered pair Γ � (G, σ), where G � (V, E) is a simple unsigned graph called the underlying graph of Γ, and σ: E ⟶ +, − { }, called a signing (or a signature), is a sign function from the edge set E to the set +, − { } of signs. e sign of a signed graph is defined as the product of signs of its edges. A signed graph is said to be positive (respectively, negative) if its sign is positive (respectively, negative), i.e., it contains an even (respectively, an odd) number of negative edges. A signed graph is said to be allpositive (respectively, all-negative) if all its edges are positive (respectively, negative), denoted by (G, +) (respectively(G, − )). A signed graph is said to be balanced if each of its cycles is positive, otherwise unbalanced.
Let A(G) � (a ij ) n×n be adjacency matrix of unsigned graph G, where a ij � 1 whenever vertices i and j are adjacent and a ij � 0 otherwise. And, let D(G) be its diagonal matrix of vertices degrees; then, L(G) � D(G) − A(G) and Q(G) � D(G) + A(G) are called Laplacian and signless Laplacian matrix of G, respectively.
Similarly, for a signed graph Γ � (G, σ), its adjacency matrix A(Γ) � (a ij ) n×n is defined as a ij � σ ij , if vertices i and j are adjacent, Suppose Γ � (G, σ) is a signed graph and θ: V ⟶ +1, − 1 { } is any sign function. Switching Γ by θ means constructing a new signed graph Γ θ � (G, σ θ ) whose underlying graph is also G, while sign function specified on Let Γ 1 � (G, σ 1 ) and Γ 2 � (G, σ 2 ) be two signed graphs on the same underlying graph G. We call Γ 1 and Γ 2 are switching equivalent; write Γ 1 ∼ Γ 2 , if there exists a switching function θ such that Γ 2 � Γ θ 1 . Actually, switching equivalent signed graphs can be considered as switching isomorphic, and their signatures are thought to be equivalent. Switching leaves many signed-graphic invariant, such as the set of positive cycles.
Two matrices M 1 and M 2 of order n are called signature similar if there exists a signature matrix, i.e., a diagonal matrix S � diag s 1 , . . . , s n with diagonal entries s i � ± 1 such that M 2 � SM 1 S. Notice that two signature similar matrices have the same eigenvalues. From the definitions of switching equivalent and signature similar, we have the following. Lemma 1. Suppose Γ 1 � (G, σ 1 ) and Γ 2 � (G, σ 2 ) be two signed graphs on the same underlying graph G; then, Γ 1 ∼ Γ 2 if and only if L(Γ 1 ) and L(Γ 2 ) are signature similar.
Signed graphs were first introduced by HARARY, in [1], in connection with the study of the theory of social balance in social psychology, see [2]. After that, there are many works on signed graph, see [3][4][5]. e energy of unsigned graph G [6][7][8] is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. e Laplacian energy of a graph G [9] is defined as For its basic properties, see [10][11][12][13][14][15]. e signless Laplacian energy of a graph G is defined as QE e Laplacian energy of a signed graph Γ, see [16], is defined similarly as are the eigenvalues of Laplacian matrix of Γ and d(Γ) is the average degree of Γ, which is the same as that of G, so we will denote it by d simply.
Two (signed) graphs of the same order are said to be Laplacian equienergetic if they have the same Laplacian energy. Two (signed) graphs are said to be Laplacian cospectral if they have the same Laplacian eigenvalues. From Lemma 1, we know that switching equivalent signed graphs must be Laplacian cospectral and Laplacian equienergetic.
Finding Laplacian equienergetic graphs is interesting, and there are so many research works on Laplacian equienergetic unsigned graphs already, see [17][18][19][20], for example, but it is very few on signed ones. In this paper, we present several infinite families of Laplacian equienergetic signed graphs which are not Laplacian cospectral.

Main Results
It seems more difficult to find Laplacian equienergetic signed graphs than unsigned ones, since we know too little about Laplacian spectrum of signed graphs. It is known that an unsigned graph could be considered as an all-positive signed graph; therefore, two Laplacian equienergetic unsigned graphs is also a pair of Laplacian equienergetic signed ones. And, we will not deal with this case.
In the paper, we try to find Laplacian equienergetic signed graph pairs with common underlying graph. at is, we are interested in graphs G which satisfy LE(G, σ 1 ) � LE(G, σ 2 ). At first, we consider graphs which satisfy LE(G, +) � LE(G, − ); in other words, we consider graphs which satisfy LE(G) � QE(G). In fact, there are such graphs trivially. It is well known that, for a regular graph, LE(G) � QE(G) holds. Besides, for bipartite graphs, the Laplacian spectrum coincides with the signless Laplacian spectrum; obviously, in this case, we have LE(G) � QE(G) also. Now, we present two infinite families of connected nonregular and nonbipartite such graphs.
Let G 1 ▽G 2 denote the join of graphs G 1 and G 2 , obtained from the union of G 1 and G 2 by joining every vertex of G 1 with every vertex of G 2 . e following lemma is from [21].

Lemma 2.
Let G 1 and G 2 be graphs on n 1 and n 2 vertices, respectively. Let L 1 and L 2 be the Laplacian matrices for G 1 and G 2 , respectively, and let L be the Laplacian matrix for e first family of such graphs is from [22].
e second family of such graphs is as follows.

□
In the sequel, we consider graphs satisfying LE(G, σ) � LE(G, +). We should point out that, by the definition of Laplacian energy of a (signed) graph, when computing its Laplacian energy, it suffices to know its Laplacian eigenvalues greater than its average degree.

Lemma 5.
Suppose Γ has exactly k Laplacian eigenvalues greater than average degree d, that is, λ 1 Besides, if two signed graphs Γ 1 � (G, σ 1 ) and Γ 2 � (G, σ 2 ) both have k Laplacian eigenvalues greater than d and the same sum of first k-greatest Laplacian eigenvalues, then Γ 1 and Γ 2 are Laplacian equienergetic.

Theorem 7
(1) For each even integer n � 4k where H k− 1 is an arbitrary graph on k − 1 vertices (connected or disconnected); the sign function σ specifies exactly 3k edges to be negative, where k edges connecting the ith vertex of K 3 to k distinct isolated vertices K 1 s, i � 1, 2, 3, and all remaining edges are positive; then, G satisfies LE(G, σ) � LE(G, +).
where H k is an arbitrary graph on k vertices (connected or disconnected), the sign function σ specifies 3k edges to be negative, where k − 1 edges connect the ith vertex of K 3 to k − 1 distinct isolated vertices K 1 s, i � 1, 2, 3, the 3 edges of K 3 . And, all remaining edges are positive; then, G satisfies LE(G, σ) � LE(G, +).
eir Laplacian spectra are shown in Table 1.

Data Availability
No data were used to support the findings of the study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.