Book Embedding of Infinite Family ((2h+3 2))-Crossing-Critical Graphs for h=1 with Rational Average Degree r∈(3.5,4)

A principal tool used in construction of crossing-critical graphs are tiles. In the tile concept, tiles can be arranged by gluing one tile to another in a linear or circular fashion. The series of tiles with circular fashion form an infinite graph family. In this way, the intersection number of this family of graphs can be determined. In this research, has been formed an infinite family graphs Q_((1,s,b) ) (n) with average degree r between 3.5 and 4. The graph formed by gluing together many copies of the tile P_((1,s,b) ) in circular fashion, where the tile P_((1,s,b) ) consist of two identical pieces of tile. And then, the graph embedded into the book to determine the pagenumber that can be formed. When embed graph into book, the vertices are put on a line called the spine and the edges are put on half-planes called the pages. The results obtained show that the graph Q_((1,s,b) ) (n) has 10-crossing-critical and book embedding of graph has 4-page book.

On Cayley graphs of {\bb Z}^4

The generating sets of {\bb Z}^4 have been enumerated which consist of integral four-dimensional vectors with components −1, 0, 1 and allow Cayley graphs without edge intersections in a straight-edge embedding in a four-dimensional Euclidean space. Owing to computational restrictions the valency of enumerated graphs has been fixed to 10. Up to isomorphism 58 graphs have been found and characterized by coordination sequences, shortest cycles and automorphism groups. To compute automorphism groups, a novel strategy is introduced that is based on determining vertex stabilizers from the automorphism group of a sufficiently large finite ball cut out from an infinite graph. Six exceptional, rather `dense' graphs have been identified which are locally isomorphic to a five-dimensional cubic lattice within a ball of radius 10. They could be built by either interconnecting interpenetrated three- or four-dimensional cubic lattices and therefore necessarily contain Hopf links between quadrangular cycles. As a consequence, a local combinatorial isomorphism does not extend to a local isotopy.

Chain conditions for graph C*-algebras

In this article, we study chain conditions for graph C*-algebras. We show that there are infinitely many mutually non isomorphic Noetherian (and Artinian) purely infinite graph C*-algebras with infinitely many ideals. We prove that if E is a graph, then {C^{*}(E)} is a Noetherian (resp. Artinian) C*-algebra if and only if E satisfies condition (K) and each ascending (resp. descending) sequence of admissible pairs of E stabilizes.

On the Number of Shortest Weighted Paths in a Triangular Grid

Counting the number of shortest paths in various graphs is an important and interesting combinatorial problem, especially in weighted graphs with various applications. We consider a specific infinite graph here, namely the honeycomb grid. Changing to its dual, the triangular grid, paths between triangle pixels (we abbreviate this term to trixels) are counted. The number of shortest weighted paths between any two trixels of the triangular grid is discussed. For each trixel, there are three different types of neighbor trixels, 1-, 2- and 3-neighbours, depending the Euclidean distance of their midpoints. When considering weighted distances, the positive values α, β and γ are assigned to the ‘steps’ to various neighbors. We gave formulae for the number of shortest weighted paths between any two trixels in various cases by the respective weight values. The results are nicely connected to various numbers well-known in combinatorics, e.g., to binomial coefficients and Fibonacci numbers.

A STRUCTURED INVERSE SPECTRUM PROBLEM FOR INFINITE GRAPHS AND UNBOUNDED OPERATORS

Let $G$ be an infinite graph on countably many vertices and let $\unicode[STIX]{x1D6EC}$ be a closed, infinite set of real numbers. We establish the existence of an unbounded self-adjoint operator whose graph is $G$ and whose spectrum is $\unicode[STIX]{x1D6EC}$.

Cuntz Splice invariance for purely infinite graph algebras

We show that the Cuntz Splice preserves the stable isomorphism class of a purely infinite graph $\mathrm{C}^*$-algebra with finitely many ideals.