On Two Laplacian Matrices for Skew Gain Graphs

Let $G=(V,\overrightarrow{E})$ be a graph with some prescribed orientation for the edges and $\Gamma$ be an arbitrary group. If $f\in \mathrm{Inv}(\Gamma)$ be an anti-involution then the skew gain graph $\Phi_f=(G,\Gamma,\varphi,f)$ is such that the skew gain function $\varphi:\overrightarrow{E}\rightarrow \Gamma$ satisfies $\varphi(\overrightarrow{vu})=f(\varphi(\overrightarrow{uv}))$. In this paper, we study two different types, Laplacian and $g$-Laplacian matrices for a skew gain graph where the skew gains are taken from the multiplicative group $F^\times$ of a field $F$ of characteristic zero. Defining incidence matrix, we also prove the matrix tree theorem for skew gain graphs in the case of the $g$-Laplacian matrix.

A gain graph is a graph with some orientation for the edges such that each edge has a gain, that is a label from a group so that reversing the direction of edge inverts the gain [9]. Generalizing the notion of gain graphs, skew gain graphs are defined such that gain of an edge (we call it as skew gain) is related to the skew gain of the reverse edge by an anti-involution [4]. The general expression for computing the coefficients of the characteristic polynomial of the adjacency matrix of skew gain graphs are studied in [8]. Laplacian matrix of a graph and matrix tree theorem are well studied by many which can be referred to for instance from [6]. The matrix tree theorem for signed graph can be seen in Zaslavsky [10] and on a more general setting in Chaiken [3]. In this paper, we define two different Laplacian matrices for skew gain graphs and prove matrix tree theorem for skew gain graphs.
Let Γ be an arbitrary group. A function f : Γ → Γ is an involution if f (f (x)) = x for all x ∈ Γ . A function f : Γ → Γ is called an anti-homomorphism if f (xy) = f (y)f (x) for all x, y ∈ Γ . For an abelian group an anti-homomorphism is always a homomorphism. An involution f : Γ → Γ which is an anti-homomorphism is called an anti-involution. We use Inv(Γ) to denote the set of all anti-involutions on Γ.
We define g : Γ → Γ such that g(x) = xf (x) for all x ∈ Γ. Definition 1.1 ([4]). Let G = (V, − → E ) be a graph with some prescribed orientation for the edges and Γ be an arbitrary group. If f ∈ Inv(Γ) be an anti-involution then the skew-gain graph Φ f = (G, Γ, ϕ, f ) is such that the skew gain function The adjacency matrix of a skew gain graph is defined when the skew gains are taken from the multiplicative group F × where F is a field of characteristic zero.
Here f ∈ Inv(Γ) is an involutive automorphism. We use the notation u ∼ v when the vertices u and v are adjacent and similar notation for the incidence of an edge on a vertex.
. Given a skew gain graph Φ f = (G, F × , ϕ, f ) its adjacency matrix A(Φ f ) = (a ij ) n is defined as the square matrix of order n = |V (G)| where In the following sections, we define Laplacian matrix and g -Laplacian matrix of skew gain graphs by defining the corresponding degree and g -degree matrices. We also define the incidence matrix of a skew gain graph.
2 Laplacian matrix for skew gain graphs We define the Laplacian charactersitic polynomial of the skew gain graph Φ f as det(xI − L(Φ f )). The eigenvalues of the Laplacian matrix, counting the multiplicities, of a skew gain graph are called the Laplacian eigenvalues or Laplacian spectra of that skew gain graph.
be an n × n matrix. Then determinant of A has the expansion det(A) = sgn(π)a 1,π(1) a 2,π(2) . . . a n,π(n) where the summation is over all permutations π on the set {1,2,3, . . . , n} and sgn(π) is the sign of the permutation π. If π is an even cycle, then sgn(π) = −1 and if π is an odd cycle, then sgn(π) = +1. Thus the sign of an arbitrary permu-tation π is (−1) Ne , where N e is the number of even cycles in cyclic representation of π.
Let L(G) denotes set of all elementary subgraphs L of G (of all orders) and graph of order n, then Proof. Let d i denotes the degree of the vertex v i and let the adjacency matrix of 0 a 1,2 a 1,3 . . . a 1,n a 2,1 0 a 2,3 . . . a 2,n . . . a n,1 a n,2 a n,3 . . . 0 Then Laplacian characteristic polynomial of skew gain graph Φ f is . . a n,1 a n,2 a n,3 . . .
Using Lemma 2.3, corresponding to the identity permutation, we get the term ). Now, for any non-identity permutation π, consider the term sgn(π)a 1,π(1) a 2,π(2) . . . a n,π(n) . Any permutation π can be expressed as a product of disjoint cycles. Thus if π fixes the i th element, a i,i = x − d(v i ). Now a cycle (ij) of length two in π corresponds to a i,j .a j,i which corresponds to the edges Thus, corresponding to the non-identity permutation π we get an elementay subgraph L of G and a 1,π(1) a 2,π(2) . . . a n,π(n) becomes where N e is the number of even cycles in π, which is same as the number of components in L having even order.
When the underlying graph of Φ f = (G, F × , ϕ, f ) is a cycle or path, we call it as a skew gain cycle or skew gain path respectively.
is a skew gain path, then its Laplacian characteristic polynomial is Corollary 2.6. If Φ f = (C n , F × , ϕ, f ) is a skew gain cycle, then its Laplacian characteristic polynomial is Proof. The only elementary subgraph L ∈ L(C n ) containing cycle as a component is All other elementary subgraphs contains K 2 as components which can be considered as matchings in C n and hence using theorem 2.4 we get is a skew gain graph with underlying graph as the star K 1,n , then its Laplacian characteristic polynomial is Now we move to the Laplacian spectra of some particular classes of skew gain graphs.
Hence by Theorem 2.8, since Φ f is regular with degree m, we get the eigenvalues Theorem 2. 10. Let Φ f = (G, F × , ϕ, f ) be a skew gain graph where G = K 1,n is a star of order n + 1.
Proof. When we put x = 0 in the characteristic polynomial of L(Φ f ) , in Corollary 2.7, we get the constant term in the polynomial as .
From this the Laplacian spectrum of Φ f becomes  3 g -Laplacian matrix for skew gain graphs a for a ∈ F belongs to the algebraic closure of The incidence matrix for a skew gain graph Φ f can be defined as follows Clearly the definitions of Laplacian, g -Laplacian and incidence matrix of a skew gain graph coincide with the corresponding definitions for ordinary graphs, signed graphs and gain graphs which are extensively studied in [1,6,7,10]. Now we define a matrix operation for the incidence matrix H(Φ f ) as follows: H # is the transpose of the matrix obtained by replacing each column element as under: (i) g(ϕ( − → e j )) replaced by ( g(ϕ( − → e j ))) −1 and η .
) . In both cases, the (i, j) th entry of H(Φ f )H # (Φ f ) coincides with the (i, j) th entry of L g (Φ f ) and hence the proof.

Now considering the matrix H # ,
Finding its determinant in a similiar way we get Theorem 3.8. If Φ f = (G, F × , ϕ, f ) is a skew gain graph of order n where G is a unicyclic graph with unique cycle C then Proof. Similiarly we get A 1 -tree is a connected unicyclic graph and a 1 -forest is a disjoint union of 1 -trees. A spanning subgraph of G which is a 1 -forest is called as an essential spanning subgraph of G . We denote the collection of all essential spanning subgraphs of G by E(G) Theorem 3.9. If Φ f = (G, F × , ϕ, f ) is a skew gain graph where G is a 1 -forest, where the product runs over all component 1 -trees Ψ having unique cycle C Ψ .
Proof. By suitable reordering of vertices and edges, if necessary, we can make the matrix L g (Φ f ) as a block diagonal matrix where the blocks corresponds to the 1tree components of the 1 -forest. Then, by Lemma 3.5, determinant det(L g (Φ f )) =

Ψ∈E(G)
det(L g (Ψ)). Now by applying Theorem 3.8 we get Now we can prove the matrix -tree theorem for skew gain graphs. Proof. Let Ψ be a spanning subgraph of Φ f having exactly n edges and let det(L g (Ψ)) = 0 . We have to prove Ψ is an essential spanning subgraph of Φ f .
That is, we have to prove that the components of Ψ are 1 -trees.
By suitable ordering of vertices and edges, we can make the matrix L g (Ψ) as a block diagonal matrix diag(A i ) where the blocks A i corresponds to the components of Ψ . Thus, det(L g (Ψ)) = If Ψ contains an isolated vertex, then the matrix L g (Ψ) has a zero row which implies det(L g (Ψ)) = 0, a contradiction.
If A i is a tree for some i, then by Theorem 3.6 we get det(L g (A i )) = 0 which implies det(L g (Ψ)) = 0, again a contradiction.
Claim: If A k is a component of Ψ then A k have same number of edges and vertices.
Suppose A k , for some k, has p vertices and p + t edges where t ≥ 1. Then the n − p vertices and n − p − t edges not in A k forms either a tree or a disconnected graph having trees as components. Both cases leads to det(L g (Ψ)) = 0, a contradiction. Hence our claim. Now all the components of Ψ have same number of edges and vertices implies the components of Ψ are 1 -trees. Hence Ψ is a spanning 1 -forest. That is Ψ is an essential spanning subgraph of Φ f . where J is a spanning subgraph of G with exactly n edges. Then by Lemma 3.10 we get det(L g (Φ f )) =

Acknowledgement
The first author would like to acknowledge her gratitude to Department of Science