We study certain quotients of generalized Artin groups which have a natural map onto D-type Artin groups, where the generalized Artin group A(T ) is defined by a signed graph T . Then we find a certain quotient G(T ) according to the graph T , which also have a natural map onto A(Dn). We prove that G(T ) is isomorphic to a semidirect product of a group K(m,n), with the Artin group A(Dn), where ...

A graph G is signed if each edge is assigned + or −. A signed graph is balanced if there is a bipartition of its vertex set such that an edge has sign − if and only if its endpoints are in different parts. The Edwards-Erdös bound states that every graph with n vertices and m edges has a balanced subgraph with at least m 2 +n−1 4 edges. In the Signed Max Cut Above Tight Lower Bound (Signed Max C...

Abstract Network embedding aims to map nodes in a network low-dimensional vector representations. Graph neural networks (GNNs) have received much attention and achieved state-of-the-art performance learning node representation. Using fundamental sociological theories (status theory balance theory) model signed networks, basing GNN on has become hot topic embedding. However, most GNNs fail use e...

Let G = (V , E) be a simple graph on vertex set V and define a function f : V → {−1,1}. The function f is a signed dominating function if for every vertex x ∈ V , the closed neighborhood of x contains more vertices with function value 1 than with −1. The signed domination number of G, γs(G), is the minimum weight of a signed dominating function on G. We give a sharp lower bound on the signed do...

Let k be a positive integer and G = (V,E) be a graph of minimum degree at least k − 1. A function f : V → {−1, 1} is called a signed k-dominating function of G if ∑ u∈NG[v] f(u) ≥ k for all v ∈ V . The signed k-domination number of G is the minimum value of ∑ v∈V f(v) taken over all signed k-dominating functions of G. The signed total k-dominating function and signed total k-domination number o...

Let G be a graph and T be a vertex subset of G with even cardinality. A T -join of G is a subset J of edges such that a vertex of G is incident with an odd number of edges in J if and only if the vertex belongs to T . Minimum T -joins have many applications in combinatorial optimizations. In this paper, we show that a minimum T -join of a connected graph G has at most |E(G)| − 1 2 |E( Ĝ )| edge...

Let C?(E) be the graph C?-algebra of a row-finite E. We give complete description vertex sets gauge-invariant regular ideals C?(E). It is shown that when E satisfies condition (L), are class which preserve (L) under quotients. That is, we show if then ideal J?C?(E) necessarily gauge-invariant. Further, ideal, it C?(E)?J?C?(F), where F (L).