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A graph is closed when its vertices have a labeling by [n] with a certain property first discovered in the study of binomial edge ideals. In this article, we explore various aspects of closed graphs, including the number of closed labelings…
A multigraph is a nonsimple graph which is permitted to have multiple edges, that is, edges that have the same end nodes. We introduce the concept of spanning simplicial complexes $\Delta_s(\mathcal{G})$ of multigraphs $\mathcal{G}$, which…
For a graph $G=(V,E)$, a set $S \subseteq V$ is a $[1,2]$-set if it is a dominating set for $G$ and each vertex $v \in V \setminus S$ is dominated by at most two vertices of $S$, i.e. $1 \leq \vert N(v) \cap S \vert \leq 2$. Moreover a set…
An integral basis of the simplest number fields of degree 3,4 and 6 over $\mathbb{Q}$ are well-known, and widely investigated. We generalize the simplest number fields to any degree, and show that an integral basis of these fields is…
We study Gibbs distributions of spins taking values in a general compact Polish space, interacting via a pair potential along the edges of a generalized random graph with a given asymptotic weight distribution $P$, obtained by annealing…
This paper investigates a generalized hyperbolic circle packing (including circles, horocycles or hypercycles) with respect to the total geodesic curvatures on the surface with boundary. We mainly focus on the existence and rigidity of…
The goal of this paper is to generalize some of the existing toolkit of combinatorial algebraic topology in order to study the homology of abstract chain complexes. We define shellability of chain complexes in a similar way as for cell…
We study the class of simple graphs $\mathcal{G}^*$ for which every pair of distinct odd cycles intersect in at most one edge. We give a structural characterization of the graphs in $\mathcal{G}^*$ and prove that every $G \in \mathcal{G}^*$…
In enumerative combinatorics, it is often a goal to enumerate both labeled and unlabeled structures of a given type. The theory of combinatorial species is a novel toolset which provides a rigorous foundation for dealing with the…
We use a construction which we call generalized cylinders to give a new proof of the fundamental theorem of hypersurface theory. It has the advantage of being very simple and the result directly extends to semi-Riemannian manifolds and to…
We define a new integer invariant of a finite graph G, the freeness index, that measures the extent to which G can be embedded in the 3-sphere so that it and its subgraphs have ``simple" complements, i.e., complements which are homeomorphic…
In this paper, we further develop the theory of circles of partition by introducing the notion of complex circles of partition. This work generalizes the classical framework, extending from subsets of the natural numbers as base sets to…
This paper establishes an upper bound on the number of generalized cospectral mates of simple graphs, where the generalized spectrum consists of the spectrum of a graph and its complement. Moving beyond the classical problem of identifying…
We analyze the space of geometrically continuous piecewise polynomial functions or splines for quadrangular and triangular patches with arbitrary topology and general rational transition maps. To define these spaces of G 1 spline functions,…
We characterize when a generalized Baumslag-Solitar group is linear, and extend the result to the fundamental groups of a graph of groups with infinite virtually cyclic vertex and edge groups.
The Stirling number of a simple graph is the number of partitions of its vertex set into a specific number of non-empty independent sets. In 2015, Engbers et al. showed that the coefficients in the normal ordering of a word $w$ in the…
The general position problem is to find the cardinality of a largest vertex subset S such that no triple of vertices of S lie on a common geodesic. For a connected graph G, the cardinality of S is denoted by gp(G) and called gp-number (or…
A class of simple graphs such as ${\cal G}$ is said to be {\it odd-girth-closed} if for any positive integer $g$ there exists a graph $G \in {\cal G}$ such that the odd-girth of $G$ is greater than or equal to $g$. An odd-girth-closed class…
A connected and nonempty graph A is defined as generalized t-edge distance-balanced, while for each edge f={\alpha}\{beta} the number of edges nearer to {\alpha} than \{beta} are equal to t-times of edges nearer to \{beta} than to {\alpha},…
We describe how to construct generalized string-net models, a class of exactly solvable lattice models that realize a large family of 2D topologically ordered phases of matter. The ground states of these models can be thought of as…