Let C(T ) be a generalized Coxeter group, which has a natural map onto one of the classical Coxeter groups, either Bn or Dn. Let CY (T ) be a natural quotient of C(T ), and if C(T ) is simply-laced (which means all the relations between the generators has order 2 or 3), CY (T ) is a generalized Coxeter group, too . Let At,n be a group which contains t Abelian groups generated by n elements. The...

A {em weak signed Roman dominating function} (WSRDF) of a graph $G$ with vertex set $V(G)$ is defined as afunction $f:V(G)rightarrow{-1,1,2}$ having the property that $sum_{xin N[v]}f(x)ge 1$ for each $vin V(G)$, where $N[v]$ is theclosed neighborhood of $v$. The weight of a WSRDF is the sum of its function values over all vertices.The weak signed Roman domination number of $G...

A signed bipartite graph is a bipartite graph in which each edge is assigned a positive or a negative sign. Let G(U, V ) be a signed bipartite graph with U = {u1, u2, · · · , up} and V = {v1, v2, · · · , vq} . Then signed degree of ui is sdeg(ui) = di = d + i − d − i , where 1 ≤ i ≤ p and d+i ( d − i ) is the number of positive(negative) edges incident with ui , and signed degree of vj is sdeg(...

It is well-known that each nonnegative integral flow of a directed graph can be decomposed into a sum of nonnegative graph circuit flows, which cannot be further decomposed into nonnegative integral sub-flows. This is equivalent to saying that indecomposable flows of graphs are those graph circuit flows. Turning from graphs to signed graphs, the indecomposable flows are much richer than that of...

We define the signed Cayley graph on Xn denoted by Sn, and study several properties such as balancing, clusterability sign-compatibility of Sn. Apart from it we also characterization canonical consistency for some n.