Let G be a graph. G is singular if and only if the adjacency matrix of graph G is singular. The adjacency matrix of graph G is singular if and only if there is at least one zero eigenvalue. The study of the singularit...Let G be a graph. G is singular if and only if the adjacency matrix of graph G is singular. The adjacency matrix of graph G is singular if and only if there is at least one zero eigenvalue. The study of the singularity of graphs is of great significance for better characterizing the properties of graphs. The following definitions are given. There are 4 paths, the starting points of the four paths are bonded into one point, and the ending point of each path is bonded to a cycle respectively, so this graph is called a kind of quadcyclic peacock graph. And in this kind of quadcyclic peacock graph assuming the number of points on the four cycles is a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>, a<sub>4</sub>, and the number of points on the four paths is s<sub>1</sub>, s<sub>2</sub>, s<sub>3</sub>, s<sub>4</sub>, respectively. This type of graph is denoted by γ (a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>, a<sub>4</sub>, s<sub>1</sub>, s<sub>2</sub>, s<sub>3</sub>, s<sub>4</sub>), called γ graph. And let γ (a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>, a<sub>4</sub>, 1, 1, 1, 1) = δ (a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>, a<sub>4</sub>), this type four cycles peacock graph called δ graph. In this paper, we give the necessary and sufficient conditions for the singularity of γ graph and δ graph.展开更多
The spectral theory of graph is an important branch of graph theory,and the main part of this theory is the connection between the spectral properties and the structural properties,characterization of the structural p...The spectral theory of graph is an important branch of graph theory,and the main part of this theory is the connection between the spectral properties and the structural properties,characterization of the structural properties of graphs.We discuss the problems about singularity,signature matrix and spectrum of mixed graphs.Without loss of generality,parallel edges and loops are permitted in mixed graphs.Let G1 and G2 be connected mixed graphs which are obtained from an underlying graph G.When G1 and G2 have the same singularity,the number of induced cycles in Gi(i=1,2)is l(l=1,l>1),the length of the smallest induced cycles is 1,2,at least 3.According to conclusions and mathematics induction,we find that the singularity of corresponding induced cycles in G1 and G2 are the same if and only if there exists a signature matrix D such that L(G2)=DTL(G1)D.D may be the product of some signature matrices.If L(G2)=D^TL(G1)D,G1 and G2 have the same spectrum.展开更多
We generalize Biggs Theorem to the case of directed cycles of multi-digraphs allowing to compute the dimension of the directed cycle space independently of the graph representation with linear runtime complexity. By c...We generalize Biggs Theorem to the case of directed cycles of multi-digraphs allowing to compute the dimension of the directed cycle space independently of the graph representation with linear runtime complexity. By considering two-dimensional CW complex of elementary cycles and deriving formulas for the Betti numbers of the associated cellular homology groups, we extend the list of representation independent topological inavariants measuring the graph structure. We prove the computation of the 2nd Betti number to be sharp #<em>P</em> hard in general and present specific representation invariant sub-fillings yielding efficiently computable homology groups. Finally, we suggest how to use the provided structural measures to shed new light on graph theoretical problems as <em>graph embeddings</em>, <em>discrete Morse theory </em>and<em> graph clustering</em>.展开更多
Let G be a finite simple graph and A(G)be its adjacency matrix.Then G is singular if A(G)is singular.The graph obtained by bonding the starting ver-tices and ending vertices of three paths Pa1,Pa2,Pa3 is calledθ-grap...Let G be a finite simple graph and A(G)be its adjacency matrix.Then G is singular if A(G)is singular.The graph obtained by bonding the starting ver-tices and ending vertices of three paths Pa1,Pa2,Pa3 is calledθ-graph,represented byθ(a1,a2,a3).The graph obtained by bonding the two end vertices of the path Ps to the vertices of theθ(a1,a2,a3)andθ(b1,b2,b3)of degree three,respectively,is denoted byα(a1,a2,a3,s,b1,b2,b3)and calledα-graph.β-graph is denoted whenβ(a1,a2,a3,b1,b2,b3)=α(a1,a2,a3,1,b1,b2,b3).In this paper,we give the necessary and sufficient conditions for the singularity ofα-graph andβ-graph,and prove that the probability that a random givenα-graph andβ-graph is a singular graph is equal to 14232048 and 733/1024,respectively.展开更多
文摘Let G be a graph. G is singular if and only if the adjacency matrix of graph G is singular. The adjacency matrix of graph G is singular if and only if there is at least one zero eigenvalue. The study of the singularity of graphs is of great significance for better characterizing the properties of graphs. The following definitions are given. There are 4 paths, the starting points of the four paths are bonded into one point, and the ending point of each path is bonded to a cycle respectively, so this graph is called a kind of quadcyclic peacock graph. And in this kind of quadcyclic peacock graph assuming the number of points on the four cycles is a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>, a<sub>4</sub>, and the number of points on the four paths is s<sub>1</sub>, s<sub>2</sub>, s<sub>3</sub>, s<sub>4</sub>, respectively. This type of graph is denoted by γ (a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>, a<sub>4</sub>, s<sub>1</sub>, s<sub>2</sub>, s<sub>3</sub>, s<sub>4</sub>), called γ graph. And let γ (a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>, a<sub>4</sub>, 1, 1, 1, 1) = δ (a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>, a<sub>4</sub>), this type four cycles peacock graph called δ graph. In this paper, we give the necessary and sufficient conditions for the singularity of γ graph and δ graph.
基金Quality Engineering Project of Anhui Province,China(No.2017zhkt036)
文摘The spectral theory of graph is an important branch of graph theory,and the main part of this theory is the connection between the spectral properties and the structural properties,characterization of the structural properties of graphs.We discuss the problems about singularity,signature matrix and spectrum of mixed graphs.Without loss of generality,parallel edges and loops are permitted in mixed graphs.Let G1 and G2 be connected mixed graphs which are obtained from an underlying graph G.When G1 and G2 have the same singularity,the number of induced cycles in Gi(i=1,2)is l(l=1,l>1),the length of the smallest induced cycles is 1,2,at least 3.According to conclusions and mathematics induction,we find that the singularity of corresponding induced cycles in G1 and G2 are the same if and only if there exists a signature matrix D such that L(G2)=DTL(G1)D.D may be the product of some signature matrices.If L(G2)=D^TL(G1)D,G1 and G2 have the same spectrum.
文摘We generalize Biggs Theorem to the case of directed cycles of multi-digraphs allowing to compute the dimension of the directed cycle space independently of the graph representation with linear runtime complexity. By considering two-dimensional CW complex of elementary cycles and deriving formulas for the Betti numbers of the associated cellular homology groups, we extend the list of representation independent topological inavariants measuring the graph structure. We prove the computation of the 2nd Betti number to be sharp #<em>P</em> hard in general and present specific representation invariant sub-fillings yielding efficiently computable homology groups. Finally, we suggest how to use the provided structural measures to shed new light on graph theoretical problems as <em>graph embeddings</em>, <em>discrete Morse theory </em>and<em> graph clustering</em>.
基金Supported by National Natural Science Foundation of China(Grant No.11561056)National Natural Science Foundation of Qinghai Provence(Grant No.2022-ZJ-924)Innovation Project of Qinghai Minzu University(Grant No.07M2022002).
文摘Let G be a finite simple graph and A(G)be its adjacency matrix.Then G is singular if A(G)is singular.The graph obtained by bonding the starting ver-tices and ending vertices of three paths Pa1,Pa2,Pa3 is calledθ-graph,represented byθ(a1,a2,a3).The graph obtained by bonding the two end vertices of the path Ps to the vertices of theθ(a1,a2,a3)andθ(b1,b2,b3)of degree three,respectively,is denoted byα(a1,a2,a3,s,b1,b2,b3)and calledα-graph.β-graph is denoted whenβ(a1,a2,a3,b1,b2,b3)=α(a1,a2,a3,1,b1,b2,b3).In this paper,we give the necessary and sufficient conditions for the singularity ofα-graph andβ-graph,and prove that the probability that a random givenα-graph andβ-graph is a singular graph is equal to 14232048 and 733/1024,respectively.
