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High dimensional, sparsely populated data spaces have been characterized in terms of ultrametric topology. This implies that there are natural, not necessarily unique, tree or hierarchy structures defined by the ultrametric topology. In…
A partial cube is a graph having an isometric embedding in a hypercube. Partial cubes are characterized by a natural equivalence relation on the edges, whose classes are called zones. The number of zones determines the minimal dimension of…
The paper explores combinatorial properties of Fibonacci words and their generalizations within the framework of combinatorics on words. These infinite sequences, measures the diversity of subwords in Fibonacci words, showing non-decreasing…
The $d$-dimensional hypercube graph $Q_d$ has as vertices all subsets of $\{1,\ldots,d\}$, and an edge between any two sets that differ in a single element. The Ruskey-Savage conjecture asserts that every matching of $Q_d$, $d\ge 2$, can be…
A connected 3-valent plane graph, whose faces are $q$- or 6-gons only, is called a {\em graph $q_n$}. We classify all graphs $4_n$, which are isometric subgraphs of a $m$-hypercube $H_m$.
In 2021, {\"O}. E\v{g}ecio\v{g}lu, V. Ir\v{s}i\v{c} introduced the concept of Fibonacci-run graph $\mathcal{R}_{n}$ as an induced subgraph of Hypercube. They conjectured that the diameter of $\mathcal{R}_{n}$ is given by…
The hyperbolic Pascal triangle ${\cal HPT}_{4,q}$ $(q\ge5)$ is a new mathematical construction, which is a geometrical generalization of Pascal's arithmetical triangle. In the present study we show that a natural pattern of rows of ${\cal…
We consider expanding maps such that the unit interval can be represented as a full symbolic shift space with bounded distortion. There are already theorems about the Hausdorff dimension for sets defined by the set of accumulation points…
As a generalization of the traditional connectivity, the g-component edge connectivity c{\lambda}g(G) of a non-complete graph G is the minimum number of edges to be deleted from the graph G such that the resulting graph has at least g…
Cut-diagrams are diagrammatic objects, defined in dimensions 1 and 2, that generalize links in 3-space and surface-links in 4-space; in dimension 1, this coincides with the theory of welded links. Using cut-diagrams, we introduce an…
A Fibonacci string is a length n binary string containing no two consecutive 1s. Fibonacci cubes (FC), Extended Fibonacci cubes (ELC) and Lucas cubes (LC) are subgraphs of hypercube defined in terms of Fibonacci strings. All these cubes…
A tree containing exactly two non-pendant vertices is called a double-star. Let $k_1$ and $k_2$ be two positive integers. The double-star with degree sequence $(k_1+1, k_2+1, 1, \ldots, 1)$ is denoted by $S_{k_1, k_2}$. If $G$ is a cubic…
Neural embeddings are a popular set of methods for representing words, phrases or text as a low dimensional vector (typically 50-500 dimensions). However, it is difficult to interpret these dimensions in a meaningful manner, and creating…
We show how to test whether a graph with n vertices and m edges is a partial cube, and if so how to find a distance-preserving embedding of the graph into a hypercube, in the near-optimal time bound O(n^2), improving previous O(nm)-time…
Structure connectivity and substructure connectivity are innovative indicators for assessing network reliability and fault tolerance. Similarly, fault diameter evaluates fault tolerance and transmission delays in networks. This paper…
This paper considers a kind of generalized measure $\lambda_s^{(h)}$ of fault tolerance in a hypercube-like graph $G_n$ which contain several well-known interconnection networks such as hypercubes, varietal hypercubes, twisted cubes,…
Perfect codes in the $n$-dimensio\-nal grid $\Lambda_n$ of the lattice $\mathbb{Z}^n$ ($0<n\in\mathbb{Z}$) and its quotient toroidal grids were obtained via the truncated distance in $\mathbb{Z}^n$ given between $u=(u_1,\cdots,u_n)$ and…
Associated to any finite metric space are a large number of objects and quantities which provide some degree of structural or geometric information about the space. In this paper we show that in the setting of subsets of weighted Hamming…
Binary embedding is a nonlinear dimension reduction methodology where high dimensional data are embedded into the Hamming cube while preserving the structure of the original space. Specifically, for an arbitrary $N$ distinct points in…
Beineke, Harary and Ringel discovered a formula for the minimum genus of a torus in which the $n$-dimensional hypercube graph can be embedded. We give a new proof of the formula by building this surface as a union of certain faces in the…