Related papers: Enumeration of balanced finite group valued functi…
An arithmetical structure on a graph is given by a labeling of the vertices which satisfies certain divisibility properties. In this note, we look at several families of graphs and attempt to give counts on the number of arithmetical…
A function on a (generally infinite) graph $\G$ with values in a field $K$ of characteristic 2 will be called {\it harmonic} if its value at every vertex of $\G$ is the sum of its values over all adjacent vertices. We consider binary…
This paper gives a key definition, for a new approach to dessins and algebraic numbers. The distant goal is to construct from each dessin $D$ an algebraic number $\eta_D$, in a systematic and useful way. The algebra of balanced dessins is…
We consider directed weighted graphs and define various families of path counting functions. Our main results are explicit formulas for the main term of the asymptotic growth rate of these counting functions, under some irrationality…
A polynomial-time exact algorithm for counting the number of directed acyclic graphs in a Markov equivalence class was recently given by Wien\"obst, Bannach, and Li\'skiewicz (AAAI 2021). In this paper, we consider the more general problem…
We study a generating function for the sum over fatgraphs with specified valences of vertices and faces, inversely weighted by the order of their symmetry group. A compact expression is found for general (i.e. non necessarily connected)…
Some of the basic properties of any dynamical system can be summarized by a graph. The dynamical systems in our theory run from maps like the logistic map to ordinary differential equations to dissipative partial differential equations. Our…
We provide algorithms involving edge slides, for a connected simple graph to evolve in a finite number of steps to another connected simple graph in a prescribed configuration, and for the regularization of such a graph by the minimization…
We study functional graphs generated by quadratic polynomials over prime fields. We introduce efficient algorithms for methodical computations and provide the values of various direct and cumulative statistical parameters of interest. These…
S. Bera (Line graph characterization of power graphs of finite nilpotent groups, \textit{Communication in Algebra}, 50(11), 4652-4668, 2022) characterized finite nilpotent groups whose power graphs and proper power graphs are line graphs.…
In a graph, we assign distinct integers to the vertices, and take the sum of two integers if they are on two adjacent vertices. The minimum possible number of different sums is the \emph{sum index} of this graph. In this paper, we present…
The sandpile group of a graph is a well-studied object that combines ideas from algebraic graph theory, group theory, dynamical systems, and statistical physics. A graph's sandpile group is part of a larger algebraic structure on the graph,…
Arakelov-Green functions defined on metrized graphs have important role in relating arithmetical problems on algebraic curves into graph theoretical problems. In this paper, we clarify the combinatorial interpretation of certain…
Using the theory of combinatorial species, we compute the cycle index for bipartite graphs, which we use to count unlabeled bipartite graphs and bipartite blocks.
Signed graphs are graphs with signed edges. They are commonly used to represent positive and negative relationships in social networks. While balance theory and clusterizable graphs deal with signed graphs to represent social interactions,…
Given a valued graph, where both the nodes and the edges of the graph are associated with one or several values, any network function for a given node must be defined in terms of that node and its connected nodes in the graph. Generally,…
We define a new weighted zeta function for a finite graph and obtain its determinant expression. This result gives the characteristic polynomial of the transition matrix of the Szegedy walk on a graph.
Aggregation functions are generally defined and used to combine several numerical values into a single one, so that the final result of the aggregation takes into account all the individual values in a given manner. Such functions are…
The solubility graph $\Gamma_S(G)$ associated with a finite group $G$ is a simple graph whose vertices are the elements of $G$, and there is an edge between two distinct vertices if and only if they generate a soluble subgroup. In this…
The $G$-graph $\Gamma(G,S)$ is a graph from the group $G$ generated by $S\subseteq G$, where the vertices are the right cosets of the cyclic subgroups $\langle s \rangle, s\in S$ with $k$-edges between two distinct cosets if there is an…