A Geometric Property of the Laplacian matrix of a Connected Nonsingular Mixed Graph

In this paper, we give a geometric interpretation of the Laplacian matrix of a connected nonsingular mixed graph which generalizes the results of M. Fiedler (M. Fiedler, Geometry of the Laplacian, Linear Algebra Appl ., 2005, 403: 409–413). In addition, the relations of geometric properties between a connected (singular or nonsingular) mixed graph, and all its resigned graphs will be characterized.


Introduction
Let G � (V, E) be a mixed graph with vertex set V � V(G) � v 1 , v 2 , . . . , v n and edge set E � E(G) � e 1 , e 2 , . . . , e m , which is obtained from an undirected simple graph by orienting some (possibly none or all) of its edges such that one of its endpoint forms a head and the another one is a tail. e incidence matrix of G is an n × m matrix M � M(G) � (m ij ) whose entries are defined by m ij � 1 if e j is an unoriented edge incident to v i or e j is an oriented edge with head v i , m ij � − 1 if e j is an oriented edge with tail v i , and m ij � 0 otherwise. e Laplacian matrix of G is defined as L(G) � (l ij ) n×n � MM T (see [1][2][3][4]), where the matrix M T denotes the transpose of M. Obviously L(G) is a Gram matrix and thus is positive semidefinite. G is called singular (or nonsingular) if L(G) is singular (or nonsingular).
A mixed graph is called quasi-bipartite if it does not contain a nonsingular cycle, or equivalently, it contains no cycles with an odd number of unoriented edges (see (1,Lemma 1)). e all-oriented graph obtained from the mixed graph G by arbitrarily orienting every unoriented edge of G (if one exists) will be denoted by G → , the signature matrix with 1 or − 1 along its diagonal of a diagonal matrix will be denoted by D. A graph is called a resigned graph of G if it is obtained from G by resigning under the signature matrix D, and that the labelling of the vertices is the same as that of G, denoted by D G.
en each resigned graph of G gives a resigning of the edges of G (that is, some oriented edges of G may turn to be unoriented and vice versa), and L( D G) � D T L(G)D. e following results are well known.
Lemma 1 (see [3], Lemma 2.2, and [5], Lemma 5). Let G be a connected mixed graph. en, G is singular if and only if G is quasi-bipartite.
Theorem 1 (see [1], eorem 4). Let G be a connected mixed graph. en, G is quasi-bipartite if and only if there exists a signature matrix D such that D T L(G)D � L( G → ). In this paper, we firstly give a geometric interpretation of the Laplacian matrix of a connected nonsingular mixed graph which generalizes the results of Fiedler [6] from undirected graphs to mixed graphs. Finally, the relations of geometric properties between a connected (singular or nonsingular) mixed graph and its resigned graphs will be characterized.

Nonsingular Mixed Graphs.
Let L � L(G) � (l ij ) n×n be the Laplacian matrix of a connected nonsingular mixed graph on n(n ≥ 3) vertices. en, rank(L) � n. By the wellknown equivalence of positive definite and Gram matrices, there exists uniquely n linear independent vectors denoted by u 1 , . . . , u n , in a Euclidean space E n of dimension n, such that the inner products of the vectors u i satisfy (1) Observe that the vectors u i and u j (i ≠ j) are orthogonal if and only if there is no edge in G between the vertices i and j. If (i, j) ∈ E(G), then the angle between u i and u j is obtuse if (i, j) is oriented, the angle is acute otherwise.
Let U � (u 1 , . . . , u n ) ∈ E n , then we have the following geometric interpretation of L(G).

Theorem 2.
Let L � L(G) � (l ij ) n×n be the Laplacian matrix of a connected nonsingular mixed graph G on n(n ≥ 3) vertices. en, there exists a unique quadric Q: and all nondiagonal entries of Z are 0, which proves that quadric equation (2) contains all the end-points of the vectors u i . e equation of the tangent hyperplane at the end-points of u k to the quadric equation (2) is Note that ( On the contrary, the hyperplane containing all vectors u i for i ≠ k satisfies the relation: us, both the hyperplanes are parallel, and equation (2) is indeed the equation of Q.
Assume there exists another quadric Q ′ : which holds properties (i) and (ii) mentioned above. en, by property (i), we have us, the equation of the tangent hyperplane at the endpoints of u k to the quadric equation (2) is Consequently, by property (ii), the equation of the hyperplane containing all the remaining n − 1 vectors is since, for each i, the vector u i contains the origin as one endpoint. Associating equations (3) and (6), we have en, B � K − 1 . Hence, the uniqueness follows. e quadric Q is an ellipsoid, usually called the Steiner circumscribed ellipsoid of the simplex with vertices in the end-points of the vector u i . Now, suppose that v is an eigenvector of L corresponding to the eigenvalue λ. We have U T Uv � λv. Consequently, UU T Uv � λUv. e vector z � Uv is thus an eigenvector of K as well as K − 1 corresponding to the same λ and is thus the direction of an axis of the quadric Q. Indeed, the halfline x � μz, μ ≥ 0, meets the quadric in the point in which the tangent is parallel to x T z � 0, thus orthogonal to z. We have thus a correspondence Γ between the eigenvector of L and the axis of Q. If the eigenvector corresponds to the eigenvalue λ, then the length of the corresponding half axis is � λ √ . Applying the method similar to that of eorem 1 in [6], we have the following. Let us transform the original orthogonal coordinate system in E n into an other orthogonal coordinate system in which the axes coincide with the axes of the quadric Q. is will be performed algebraically as follows.
e matrix K can be written as OΛO T , where O is an orthogonal matrix and Λ is a diagonal matrix of the eigenvalues of K. Since K − 1 � OΛ − 1 O T , equation (2) of Q will be transformed into where y � O T x is again the column vector of orthogonal coordinates in the transformed system. en, in the new system, the quadric Q has the equation: Journal of Chemistry L(G), and α(G) is the smallest eigenvalue of L(G). en, the radius of the externally tangent circle of Q is ���� � d(G), and the radius of the internally tangent circle of Q is ���� � α(G). en, the vector u i transforms into the vector w i � O T u i for i � 1 to n. e following result seems interesting, we say as usual that an affine transformation is such a linear transformation which transforms points at infinity into points at infinity. Theorem 4. Let A be the affine transformation in E n which transforms the ellipsoid Q into a hypersphere: en, A transforms the vector w i into vector W i � d (− 1/2) i w i which form an eutactic star, being all of equal length and spanning equal mutual angles.
Proof. By equation (14), we obtain the matrix form where Λ was defined earlier. Since w i � O T u i and W i � Λ (− 1/2) w i , we have for the matrix W whose columns are the coordinates of the vectors W i , W � Λ (− 1/2) O T U. erefore, the Gram matrix of the vectors W i is since this is the matrix Z mentioned above. Immediately, the last matrix is the Gram matrix of an eutactic star.

Mixed Graphs with the Same Underlying.
In this section, we will characterize the relations of geometric properties between the graph G (singular or nonsingular) and all its resigned graphs D G. Denoted by K − 1 (G), the matrix of quadric corresponding to the graph G, by D u i the column vector of the matrix U D since more than one graph is under discussion. Assume L � L(G) � (l ij ) n×n be the Laplacian matrix of a connected mixed graph G on n vertices. Let ℵ(G) be the set of all resigned graphs of a mixed graph G. Obviously, ℵ(G) has the following properties: (i) G ∈ ℵ(G) (ii) All elements of ℵ(G) have the same singularity as that of the graph G (iii) All elements of ℵ(G) can be considered as the where U � (u 1 , . . . , u n ) has been defined above. Consequently, we have the following.