Principal minors of Hermitian (quasi-)Laplacian matrix of second kind for mixed graphs

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Introduction
We only consider simple and finite connected graphs in this paper. Let G be an unoriented graph with the vertex set V = {v 1 , v 2 , . . . , v n } and edge set E, in which |V | = n and |E| = m. The degree d i of a vertex v i in G is the number of edges incident with v i . A mixed graph M G is obtained from an unoriented graph G by giving directions to some edges in G. So G is often referred to the underlying graph of M G . A mixed M G is called a mixed subgraph of M G if G is a subgraph of G and the direction of all the edges of M G is the same as in M G . For two adjacent vertices v i and v j of M G , we denote an oriented edge from v i to v j by v i → v j or v j ← v i . Similarly, we denote by v i ↔ v j an unoriented edge between v i and v j . Usually, we also use M C to represent a mixed cycle.
Until now the research on spectral theories of mixed graphs has been paid more and more attention. Recall that, for a mixed graph M G , its Hermitian adjacency matrix of the first kind H(M G ) = (h ij ) was proposed by Guo and Mohar [5] and independently by Liu and Li [8] and 0 otherwise. For this matrix associated to the mixed graph, some basic spectral properties were also established in [5,8]. In 2015, Yu and Qu [15] introduced the incident matrix and Hermitian Laplacian matrix of the first kind for a mixed graph M G and determined the positive of M G . Subsequently, Yu et al. [14] characterized the singularity of the Hermitian (quasi-)Laplacian matrix of the first kind for M G and gave the concise expressions of the determinants of these two matrices. In 2017, an analytical expression for the principal minors of the Hermitian (quasi-)Laplacian matrix of the first kind was derived in [16].
Recently, a Hermitian adjacency matrix of the second kind, denoted by N (M G ), for mixed graphs was proposed by Mohar [9], in which n ij is the sixth root of unity ω = 1+ is Hermitian, so its eigenvalues are all real. At the same time, Mohar [9] pointed out the necessity of studying this novel matrix and gave some basic spectral results in spectral graph theory. The main reason is that the sixth root of unity satisfies ω · ω = 1 and ω + ω = 1, which makes it more natural to study the relationship between eigenvalues and combinatorial properties. Moreover, the sixth root of unity also appears across applications, such as in the definition Eisenstein integers [10,13], Quantum Field Theory [6] and so on. In 2022, Li and Yu [7] studied the characteristic polynomial of this matrix and obtained an upper bound on its spectral radius. Using switching equivalence, they also studied properties of mixed graphs that are cospectral based on this matrix. For more details about this matrix, we refer readers to [7,9].
In this paper we give two incidence matrices of the second kind for a mixed graph M G , denoted by S(M G ) and T (M G ) hereinafter. Using these two matrices, we introduce the concept of Hermitian Laplacian matrix of the second kind (resp. Hermitian quasi-Laplacian matrix of the second kind) for a mixed graph M G , which is represented by is the degree diagonal matrix of M G . Bapat et al. [1] proposed a real Laplacian matrix for mixed graphs and obtained matrix tree theorem of mixed graphs in 1999. Inspired by this research and [16], we characterize the principal minor version of the matrix tree theorem based on the (quasi-)Hermitian Laplacian matrix of the second kind for M G . In addition, we also give the explicit expressions about the determinants of these two matrices L(M G ) and Q(M G ) for the mixed graph M G and present some necessary and sufficient conditions for the Hermitian Laplacian matrix of the second kind L(M G ) to be singular.
It is worth noting that the spectral theories of mixed graphs arising from the Hermitian matrices of first and second kind can be both embodied in the theory of gain graphs. The Hermitian matrix of the first kind for a mixed graph M G is the (Hermitian) adjacency matrix associated to the gain graph Γ obtained from G by choosing the weight e π 2 i for each oriented edge v i v j , and the weight 1 for all the unoriented edges. Similarly, the Hermitian matrix of the second kind for a mixed graph M G is the (Hermitian) adjacency matrix associated to the gain graph Γ obtained from G by choosing the weight e π 6 i for each oriented edge v i v j , and the weight 1 for all the unoriented edges. It turns out that many results in this paper could be alternatively deduced from the correspondent results involving complex unit gain graphs and quaternion unit gain graphs (see [2-4, 11, 12]). Our proofs ignore the overlapping of the two contexts; thus, no knowledge on gain graphs is required to understand them.

Hermitian Laplacian matrix of the second kind for mixed graphs
Suppose that M G is a mixed graph of order n and size m. An incidence matrix of the second kind of M G is an n × m matrix S(M G ) = (s ie ) with entries where s je is a complex number such that |s je | = 1. Notice that we say "an" incidence matrix of the second kind of M G because S(M G ) is not unique.
The following lemma is a particular case of Lemma 3.1 in [11]. Considering that the details of the proof are different, we give a complete proof below. Proof. First we let S(M G )S(M G ) * = (σ ij ) n×n . It is easy to see that σ ij = e∈E s ie · s je . And then we just need to compare the entries between two matrices L(M G ) and S(M G )S(M G ) * . We divide it into the following two cases.

Now we consider this value according to the direction of the edge between
Based on the discussion above, we obtain L( The above lemma further implies that L(M G ) is a positive semidefinite matrix. Let W = v i1 v i2 · · · v i k be a mixed walk of length k, and its weight is denoted by n(W ) = n i1i2 n i2i3 · · · n i k−1 i k , in which n ij is the element in the i-th row and j-th column of the Hermitian adjacency matrix of the second kind. It is easy to verify that, if the weight of W is λ for one given direction, then the weight is λ after the directions of all edges are reversed. Furthermore, let n(M C ) = n 12 n 23 · · · n n−1,n n n1 be the weight of a mixed cycle M C with n vertices. Next we consider four different types of mixed cycles depending on their weights. If n(M C ) = 1, then M C is called a positive cycle.
, then M C is called a semi-negative cycle. Moreover, we call a mixed graph M G to be positive whenever each mixed cycle of M G is positive. Lemma 2.2. Let M G be a mixed graph with n vertices and m edges. Then the matrix L(M G ) is singular if and only if any walk W = v i1 v i2 · · · v i k has the same weight for 1 ≤ k ≤ n. And at this time, there exists an eigenvector η = (1, n(W 12 ), n(W 13 ), . . . , n(W 1n )) such that L( Proof. First, assume that the matrix L(M G ) is singular. Then there exists a non-zero vector ξ = (ξ 1 , ξ 2 , . . . , ξ n ) such that L(M G )ξ = 0, which implies that S(M G ) * ξ = 0. Now, by some calculations, one has Hence, any walk W = v i1 v i2 · · · v i k has the same weight for 1 ≤ k ≤ n. Furthermore, This also implies that Therefore, L(M G ) is a singular matrix.
Observe that, by some careful checking, we find that Theorem 4 in [15] still holds in our discussion. Thus we have the following theorem from Theorem 4 in [15] and Lemma 2.2. Remark that we can also draw this theorem from Proposition 2.1 in [4]. The following lemma can be easily deduced from Lemma 6.7 in [3], but here we give a different proof according to the method of determinant expansion. Proof. Remark that the proof is similar to that of Theorem 6 in [14]. Without loss of generality, assume that the edge e i = v i v i+1 for 1 ≤ i ≤ n − 1 and e n = v n v 1 in the underlying graph of M C . Clearly, S(M C ) is a square matrix. Then expanding along the first row, one has and From the definition of S(M C ), one has s ie s je = −n ij for any edge e = v i v j . Thus it follows from (6) and (7)  s iei = 2 + (−1) n+1 s 1e1 s 2e1 · · · s nen s 1en + (−1) n+1 s 2e1 s 1e1 · · · s 1en s nen = 2 + (−1) n+1 (−1) n n 12 n 23 · · · n n1 + (−1) n+1 (−1) n n 21 · · · n 1n This completes the proof.
The following statement is analogous to an existing result related to the Hermitian Laplacian matrix of the first kind of mixed graphs (see Theorem 5 in [14]), and it can also be obtained from Proposition 2.1 in [4]. Combining the proof of Theorem 3.1 in [2] with Lemma 2.2 in [12], we can easily get the following conclusion, so we ignore this proof here. Next we turn our attention to the principal minor version of matrix tree theorem in Hermitian Laplacian matrix of the second kind for mixed graphs. Before that, we introduce some definitions, which are similar to the ones in [1]. A subgraph Γ of a connected mixed graph M G is called essential Laplacian spanning subgraph of M G if M G is positive and Γ is a spanning tree of its underlying graph G; or M G is not positive, Γ has the same vertices as M G , its components are unicyclic mixed graphs and the Hermitian Laplacian matrix of the second kind of each cycle is non-singular. An r-reduced Laplacian spanning substructure R of M G is defined as follows: R is a subgraph of M G with n − r vertices and no Laplacian singular cycles, and the number of vertices and edges in each component of R is the same. It is clear that each component of R is either a rootless tree or a Laplacian non-singular unicyclic graph. Furthermore, we can establish a one-to-one correspondence between R and Γ, in which one vertex is deleted in each positive component of Γ.
The following two lemmas and their proofs are similar to Lemma 2 and Theorem 1 in [16], we ignore some details here.
Lemma 2.4 (see [16]). For a mixed graph M G , if a substructure R of M G is a rootless tree, then | det S(R)| = 1.
where E n−r is a subset of E with n − r edges and R is the substructure with the pair (V n−r , E n−r ).
In the following, we consider the contribution of a substructure R to R | det S(R)| 2 . Since each component of R is either a rootless tree, or a Laplacian non-singular unicyclic graph. Thus we will discuss the following two cases. , then det L(R ) = 3. Based on the discussion above, we get that a rootless tree contributes 1, and each non-singular unicyclic graph contributes 1, 3 or 4 to the value of determinant. Hence, the required result follows.
where q τ1,τ2 is the number of essential Laplacian spanning subgraphs which contain τ 1 semi-negative cycles, τ 2 negative cycles and and we stipulate that q 0,0 = 0.
Proof. If M G is a mixed tree, then the result holds by Theorem 2.2. Next assume that M G contains mixed cycles. Then the Cauchy-Binet Theorem implies that L( It can be verified that all the subgraphs with vertex set V and edge set E belong to elementary Laplacian spanning subgraph Γ. Hence the required result follows by Theorem 2.4.

Hermitian quasi-Laplacian matrix of the second kind for mixed graphs
Suppose that M G is a mixed graph of order n and size m. A quasi-incidence matrix of the second kind of M G is an n × m matrix T (M G ) = (t ie ) with entries where t je is a complex number such that |t je | = 1. Notice that we say "an" quasi-incidence matrix of the second kind of M G as T (M G ) is not unique. Similar to the proof of Lemma 2.1, we obtain the following lemma. Proof. First assume that the matrix Q(M G ) is singular. Then there exists a non-zero vector ξ = (ξ 1 , ξ 2 , . . . , ξ n ) such that Q(M G )ξ = 0, which implies that T (M G ) * ξ = 0. By some calculations, one obtains Since T (M G ) * ξ = 0, then ξ i = −n ij ξ j for each edge e = v i v j of M G . It follows that implying that the result holds.
Conversely, assume that any two walks W = v i1 v i2 · · · v i k with the same parity have the same weight for 1 ≤ k ≤ n. Now let ζ = (ζ 1 , ζ 2 , . . . , ζ n ) be a vector such that ζ i = −n ij ζ j for each edge e = v i v j . Thus, it is easy to verify that Therefore, Q(M G ) is a singular matrix. Proof. Similar to the proof of Lemma 2.3, we can get The definition of T (M C ) implies that t ie t je = n ij for any edge e = v i v j . So we have · · t nen t 1en + (−1) n+1 t 2e1 t 1e1 · · · t 1en t nen = 2 + (−1) n+1 n 12 n 23 · · · n n1 + (−1) n+1 n 21 · · · n 1n This completes the proof. Proof. By the same argument as the proof of Theorem 2.3, we get the required result.
Next we consider the principal minor of Hermitian quasi-Laplacian matrix of the second kind for mixed graphs. First we introduce some definitions. A subgraph Γ of a connected mixed graph M G is called essential quasi-Laplacian spanning subgraph of M G if M G is positive and Γ is a spanning tree of its underlying graph G; or M G is not positive, Γ has the same vertices as M G , its components are unicyclic mixed graphs and the Hermitian quasi-Laplacian matrix of the second kind of each cycle is non-singular. An r-reduced quasi-Laplacian spanning substructure R of M G is defined as follows: R is a subgraph of M G with n − r vertices and no quasi-Laplacian singular cycles, and the number of vertices and edges in each component of R is the same. It is clear that each component of R is either a rootless tree or a quasi-Laplacian non-singular unicyclic graph. Furthermore, we can establish a one-to-one correspondence between R and Γ, in which one vertex is deleted in each positive component of Γ.
where E n−r is a subset of E with n − r edges and R is the substructure with the pair (V n−r , E n−r ).
In the following, we consider the contribution of a substructure R to R | det T (R)| 2 . Since each component of R is either a rootless tree, or a quasi-Laplacian non-singular unicyclic graph, so we will discuss the following two cases.

1,
if n is odd.
According to the discussion above, we get that a rootless tree contributes 1, and each non-singular unicyclic graph contributes 1, 3 or 4 to the value of determinant. It follows from (11) that the result holds.
The following statement is a direct result of Theorem 3.3, and we ignore its proof. where q τ1,τ2,τ3,τ4 is the number of essential quasi-Laplacian spanning subgraphs which contain τ 1 semi-positive odd cycles, τ 2 semi-negative even cycles, τ 3 positive odd cycles, τ 4 negative even cycles and we stipulate that q 0,0,0,0 = 0.