An assignment of positive integer weights to the edges of a simple graph G is called irregular, if the weighted degrees of the vertices are all different. The irregularity strength, s(G), is the maximal weight, minimized over all irregular assignments. In this study, we show that s(G) c1 n / , for graphs with maximum degree n and minimum

A graph is locally irregular if every two adjacent vertices have distinct degrees. Recently, Baudon et al. introduced the notion of decomposition into locally irregular subgraphs. They conjectured that for almost every graphG, there exists a minimum integer χirr(G) such thatG admits an edge-partition into χ ′ irr(G) classes, each of which induces a locally irregular graph. In particular, they c...

A closed walk in a connected graph G that contains every edge of G exactly once is an Eulerian circuit. A graph is Eulerian if it contains an Eulerian circuit. It is well known that a connected graph G is Eulerian if and only if every vertex of G is even. An Eulerian walk in a connected graph G is a closed walk that contains every edge of G at least once, while an irregular Eulerian walk in G i...

For a simple graph G, a vertex labeling φ : V (G) → {1, 2, · · · , k} is called k-labeling. The weight of an edge xy in G, denoted by wπ(xy), is the sum of the labels of end vertices x and y, i.e. wφ(xy) = φ(x) + φ(y). A vertex k-labeling is defined to be an edge irregular k-labeling of the graph G if for every two different edges e and f , there is wφ(e) 6= wφ(f). The minimum k for which the g...

A graph G is pluperfect if one can add new multiple edges to obtain a multi graph M satisfying: (1) G spans M, (2) the degrees of the nodes of M are consecutive integers, (3) G and M have the same minimum degree. Our purpose is to display several families of pluperfect graphs in order to L Introduction My good friends Mehdi Behzad and Gary Chartrand wrote a note [1] titled "No graph is perfect"...

A deBruijn sequence of order k, or a k-deBruijn sequence, over an alphabet A is a sequence of length |A| in which the last element is considered adjacent to the first and every possible k-tuple from A appears exactly once as a string of k-consecutive elements in the sequence. We will say that a cyclic sequence is deBruijn-like if for some k, each of the consecutive k-element substrings is disti...

In this paper, we introduce a generalized value iteration network (GVIN), which is an end-to-end neural network planning module. GVIN emulates the value iteration algorithm by using a novel graph convolution operator, which enables GVIN to learn and plan on irregular spatial graphs. We propose three novel differentiable kernels as graph convolution operators and show that the embedding-based ke...

This paper presents GEMS (Graph database Engine for Multithreaded Systems), a software infrastructure that enables large-scale, graph databases on commodity clusters. Unlike current approaches, GEMS implements and explores graph databases by primarily employing graph-based methods. This is reflected in all the layers of the software stack. On one hand, this allows exploiting the space efficienc...

A graph theoretical analogue of Brauer-Siegel theory for zeta functions of number …elds is developed using the theory of Artin L-functions for Galois coverings of graphs from parts I and II. In the process, we discuss possible versions of the Riemann hypothesis for the Ihara zeta function of an irregular graph.