A {\it graph product} $G$ on a $\Gamma$ is group defined as follows: For each vertex $v$ of there corresponding non-trivial $G_v$. The the quotient free product $G_v$ by commutation relations $[G_v,G_w]=1$ for all adjacent and $w$ in $\Gamma$. finitely presented has semistable fundamental at $\infty$} if some (equivalently any) finite connected CW-complex $X$ with $\pi_1(X)=G$, universal cover ...

In this work we prove that, given a simplicial graph Γ $\Gamma$ and family G $\mathcal {G}$ of linear groups over domain R $R$ , the product $\Gamma \mathcal is [ t ̲ ] $R[\underline{t}]$ where $\underline{t}$ tuple finitely many linearly independent variables. As consequence, obtain that any complex numbers again group numbers. This solves an open problem Hsu Wise (Michigan Math. J. 46 (1999), ...

The intersection graph of a family of arcs on a circle is called a circular-arc graph. This class of graphs admits some interesting subclasses: proper circular-arc graphs, unit circular-arc graphs, Helly circular-arc graphs and clique-Helly circular-arc graphs. The intersection graph of a family of chords in a circle is called a circle graph. Analogously, this class of graphs admits some subcla...

In this paper, we investigate the heat kernel embedding as a route to graph representation. The heat kernel of the graph encapsulates information concerning the distribution of path lengths and, hence, node affinities on the graph; and is found by exponentiating the Laplacian eigen-system over time. A Young–Householder decomposition is performed on the heat kernel to obtain the matrix of the em...

In this paper, we study the relationship between the Asymmetric Traveling Salesman Problem (ATSP) and the Cycle Cover Problem in terms of the strength of the triangle inequality on the edge costs in the given complete directed graph instance, G = (V, E). The strength of the triangle inequality is captured by parametrizing the triangle inequality as follows. A complete directed graph G = (V, E) ...

An interval graph is the intersection graph of a finite set of intervals on a line and a circular-arc graph is the intersection graph of a finite set of arcs on a circle. While a forbidden induced subgraph characterization of interval graphs was found fifty years ago, finding an analogous characterization for circular-arc graphs is a long-standing open problem. In this work, we study the inters...