Related papers: Compositions of Graphs Revisited

A \emph{composition} is a sequence of positive integers, called \emph{parts}, having a fixed sum. By an \emph{$m$-congruence succession}, we will mean a pair of adjacent parts $x$ and $y$ within a composition such that $x\equiv y(\text{mod}…

We present the basic ideas of forms (a generalization of Ehresmann's sketches) and their theories and models, more explicitly than in previous expositions. Forms provide the ability to specify mathematical structures and data types in any…

Algorithms to generate various combinatorial structures find tremendous importance in computer science. In this paper, we begin by reviewing an algorithm proposed by Rohl that generates all unique permutations of a list of elements which…

Assume that $G$ is a finite group. For every $a, b \in\mathbb N,$ we define a graph $\Gamma_{a,b}(G)$ whose vertices correspond to the elements of $G^a\cup G^b$ and in which two tuples $(x_1,\dots,x_a)$ and $(y_1,\dots,y_b)$ are adjacent if…

In 1963 [Ann. of Math. {\bf 78}, 267-288], Gerstenhaber invented a \emph{comp(osition)} calculus in the Hochschild complex of an associative algebra. In this paper, the first steps of the Gerstenhaber theory are exposed in an abstract (comp…

We consider sequences of integers defined by a system of linear inequalities with integer coefficients. We show that when the constraints are strong enough to guarantee that all the entries are nonnegative, the generating function for the…

In this survey we overview known results and get several new results on digraph compositions which generalize several classes of digraphs, such as quasi-transitive digraphs. After an introductory section, the paper is divided into six…

Motivated by a recent extension of the zero-one law by Kolaitis and Kopparty, we study the distribution of the number of copies of a fixed disconnected graph in the random graph $G(n,p)$. We use an idea of graph decompositions to give a…

We extend the concept of the law of a finite graph to graphings, which are, in general, infinite graphs whose vertices are equipped with the structure of a probability space. By doing this, we obtain a vast array of new unimodular measures.…

Graphs are used to represent and analyze data in domains as diverse as physics, biology, chemistry, planetary science, and the social sciences. Across domains, random graph models relate generative processes to expected graph properties,…

In this talk, we are concerned with the formulation and understanding of the combinatorics of time-ordered n-point functions in terms of the Hopf algebra of field operators. Mathematically, this problem can be formulated as one in…

Let A be any set of positive integers and n a positive integer. A composition of n with parts in A is an ordered collection of one or more elements in A whose sum is n. We derive generating functions for the number of compositions of n with…

Generalizing the notion of staircase words, introduced by Knopfmacher et.\ al, we define staircase graph words. These are functions $w$ from the vertex set $V$ of a graph into the set $\{1,2,\ldots,k\}$, such that $|w(x)-w(y)|\leq 1$, for…

This paper introduces a notion of presentation for locally inverse semigroups and develops a graph structure to describe the elements of locally inverse semigroups given by these presentations. These graphs will have a role similar to the…

Expander graphs, due to their mixing properties, are useful in many algorithms and combinatorial constructions. One can produce an expander graph with high probability by taking a random graph (e.g., the union of $d$ random bijections for a…

This paper studies infinite graphs produced from a natural unfolding operation applied to finite graphs. Graphs produced via such operations are of finite degree and automatic over the unary alphabet (that is, they can be described by…

We consider compositions of natural numbers when there are different types of each natural number. Several recursions as well as some closed formulas for the number of compositions is derived. We also find its relationships with some known…

In this paper, we explore the concept of the ``matrix product of graphs," initially introduced by Prasad, Sudhakara, Sujatha, and M. Vinay. This operation involves the multiplication of adjacency matrices of two graphs with assigned labels,…

Compositional generalization is the capacity to recognize and imagine a large amount of novel combinations from known components. It is a key in human intelligence, but current neural networks generally lack such ability. This report…

We introduce coloring groups, which are permutation groups obtained from a proper edge coloring of a graph. These groups generalize the generalized toggle groups of Striker (which themselves generalize the toggle groups introduced by…