A combinatorial description of the determinant of the Laplacian matrix of weighted directed graphs is provided here. The Laplacian spectrum of the class of connected 3-colored digraphs containing exactly one nonsingular cycle is studied here. If the weights of the edges in G are ±1, then our definition of L(G) coincides with the Laplacian matrix of a mixed graph as defined in (we show the edges with weight 1 as directed and the edges with weight −1 as not -corrected). [4].
We show that the singularity of the Laplacian matrix of weighted directed graphs is closely related to the structure of the graph. We provide a combinatorial description of the determinant of the Laplacian matrix of weighted directed graphs related to the graph structure. In Section 2.1 we provide some characterizations of the singularity of the Laplacian matrix of weighted directed graphs.
In Section 2.4, we establish a relationship between the determinant of the Laplacian matrix of weighted directed graphs and the structure of the graph. Since D∗L(G)D is Hermitian, we see that D∗L(G)D is the Laplacian matrix of the unweighted undirected graph of G. Recall that the class of 3-color digraphs contains mixed graphs, but is a subclass of subclass of weighted directed graphs.
Therefore, the structure of a connected singular 3-color digraph naturally extends the structure of an unweighted undirected graph. In this section, we describe the determinant of the Laplacian matrix of a weighted directed graph. The following lemma gives the determinant of the Laplacian cycle matrix in a weighted directed graph.
We give a complete characterization of the adjacency spectrum (ie Laplacian) of the graph resulting from such an operation. Next, we relate the Laplacian spectrum (respectively proximity) of mixed graphite G to that of G[b]. Then Gb is D-similar to a mixed graph H such that the first eigenvector of H is nonnegative at the vertices of the nonsingular cycle in H.
Unicyclic 3-colored digraph with second smallest Laplacian eigenvalue
Of all non-singular unicyclic three-color digraphs of fixed perimeter m, the smallest Laplace eigenvalue is maximized by a non-singular unicyclic three-color digraph of perimeter m with the following property: the weight of the cycle is −1 and each vertex on the cycle is only adjacent to a non-negative number of hanging vertices outside the cycle. In Chapter 2 we saw that the concept of three-color digraph is a generalization of the concept of mixed graph. The adjacency matrix of a three-color digraph distinguishes the orientations of the green edges and therefore the spectral properties of A(G) can provide additional structural information about the graph.
In Section 5.1, we provide a combinatorial interpretation of the coefficients of P(G;x) in terms of the graph structure, for a 3-colored digraph G. In Section 5.3, we provide the structure of the unicyclic 3-colored digraphs containing the weight cycle − 1 that satisfies the SR property. In Section 5.4, we provide the structure of the unicyclic 3-colored digraphs containing the cycle of weight±i satisfying the SR property.
Let G be a 3-color connected unicyclic digraph with cycle C. i) If wC = 1, then G is D-similar to a unicyclic graph with all edges red. ii) If wC =−1, then G is D-like a unicyclic graph with all edges red except for one edge in C which is blue. iii). In view of Lemma 5.1.4, unicyclic 3-color digraphs with all red edges satisfying SR properties were characterized in [7]. From now on, all our 3-color unicyclic digraphs have all edges red, except perhaps one edge at C, which is either blue or green.
Thus GisD is similar to the 3-color digraph obtained from H by adding a dependent vertex to each vertex of H with a red edge. It is natural to wonder whether a 3-color digraph with SR property is necessarily a simple corona. We say that a tree branch is dished if the order of the tree branch is odd or even, respectively.
In light of the previous observation, Γ must then be an even cycle containing alternating eis and fjs. In light of the previous observation, the maximum degree of a vertex in H is then 2. Let G be a connected non-corona unicyclic three-color digraph that satisfies the SR property and contains no green edge.
Unicyclic 3-colored digraphs with a blue edge on C
Moreover, ifg≡0 (mod 4), thenG is a simple corona. c) If G is not a simple corona, then G has exactly two odd tree branches at two different vertices of the cycle. In view of Lemma 5.3.1, let Tu and Tv be the odd tree branches at vertices u and v on C. If case 1 (resp. case 2) holds, we can extend this alternating path along p1 (resp. along p1 and p2) to obtain two alternating paths of length at least 5, starting and ending at a vertex of the cycle.
We will now examine the non-corona unicyclic graphs Gb with girth g = 6 that satisfy SR property. In that figure, Fu, Fv are forests consisting of corona trees and Tu∗, Tv∗ are trees induced by V(Fu)∪ {u, u0} and V(Fv)∪ {v, v0}, respectively. By B(α|β) we denote the submatrix of Bn×n obtained by deleting the rows and columns corresponding to α and β, respectively.
13] Let G be the graph obtained by connecting the vertex of the graph G1 to the vertex u2 of the graph G2 by an edge. Let G′i be the induced subgraph of Gi, obtained by removing the vertex ui from Gi, because i = 1,2. According to Lemma 5.2.7, Tu∗, Tv∗and the trees in Fu, Fv have SR property, because they are simple corona trees.
Then G has SR property if and only if G is a simple corona with perimeter g= 2k, k≥3 or G has the structure shown in Figure 5.3, up to D similarity.
Unicyclic 3-colored digraph with a green edge on C
Abreu, Old and new results on algebraic connectivity of graphs, Linear Algebra and its applications. Fiedler, A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory, Czechoslovak Mathematical Journal.
