In this letter, we present a linear-complexity encoding algorithm for any cycle GF(2) code CE(G,H). We just need to investigate the case where G is a nontrivial connected graph. If G is a tree, the only codeword is the all-zero word. If G is not a tree, first, we show that through graph analysis H can be transformed into an equivalent block-diagonal upper-triangular form simply by permuting the...

A graph G is (a, b)-choosable if for any color list of size a associated with each vertices, one can choose a subset of b colors such that adjacent vertices are colored with disjoint color sets. This paper shows an equivalence between the (a, b)-choosability of a graph and the (a, b)-choosability of one of its subgraphs called the extended core. As an application, this result allows to prove th...

A matchstick graph is a crossing-free unit-distance in the plane. Harborth conjectured 1981 that maximum number of edges with $n$ vertices $\lfloor 3n-\sqrt{12n-3}\rfloor$. Using Euler formula and isoperimetric inequality, it can be shown has no more than $3n-\sqrt{2\pi\sqrt{3}\cdot n}+O(1)$ edges. We improve this upper bound to $3n-c\sqrt{n-1/4}$ edges, where $c=\frac12(\sqrt{12} + \sqrt{2\pi\...

In this paper, we study a conjecture of Andries E. Brouwer from 1996 regarding the minimum number of vertices of a strongly regular graph whose removal disconnects the graph into non-singleton components. We show that strongly regular graphs constructed from copolar spaces and from the more general spaces called ∆-spaces are counterexamples to Brouwer’s Conjecture. Using J.I. Hall’s characteriz...

We introduce a method for automatically labelling edges of word co-occurrence graphs with semantic relations. Therefore we only make use of training data already contained within the graph. Starting point of this work is a graph based on word co-occurrence of the German language, which is created by applying iterated co-occurrence analysis. The edges of the graph have been partially annotated b...

Graph is a useful tool to model complicated data structures. One important task in graph analysis is assessing node proximity based on graph topology. Recently, RandomWalk with Restart (RWR) tends to pop up as a promising measure of node proximity, due to its proliferative applications in e.g. recommender systems, and image segmentation. However, the best-known algorithm for computing RWR resor...

We introduce and analyze a particular generalization of the Erdös-Rényi random graph model that is based on adding not only edges but also copies of small graphs onto the nodes of a graph. The resulting model is analytically tractable and can generate random graphs with local structures that are not tree like. First, we introduce the simplest generalization called the triplet model correspondin...

On January 18, 2022, around 1 million kilometers from Earth, five times the distance Earth to Moon, a large asteroid passed without harm Earth. Theoretically, however, event of falling into causing tsunami, is possible since there are over 27,000 near-Earth asteroids [1], and Earth’s surface covered in 71 percent by water. We introduce novel graph-grammar-based framework for tsunami simulations...

Consider a graph G with |V (G)| = p and |E(G)| q let f : V (G) → {k, k + 1, 2, . − 1}} be an injective function. The induced edge labeling ∗ for vertex is defined by (e) all e uv ∈ E(G) bijective. If f(V (G)) ∪ {f E(G)} , 1}, then called k-super cube root mean labeling. such exists, graph. In this paper, I introduce prove the existence of to graphs viz., triangular snake Tn, double D(Tn), Quadr...