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We perform a detailed computational study of the recently introduced Sombor indices on random graphs. Specifically, we apply Sombor indices on three models of random graphs: Erd\"os-R\'enyi graphs, random geometric graphs, and bipartite…
In their precedent work, the authors constructed closed oriented hyperbolic surfaces with pseudo-Anosov homeomorphisms from certain class of integral matrices. In this paper, we present a very simple algorithm to compute the Teichmueller…
Let $G=(V,E)$ be a finite simple graph. The Sombor index $SO(G)$ of $G$ is defined as $\sum_{uv\in E(G)}\sqrt{d_u^2+d_v^2}$, where $d_u$ is the degree of vertex $u$ in $G$. In this paper, we study this index for certain graphs and we…
This survey article describes the algorithmic approaches successfully used over the time to construct hyperbolic structures on 3-dimensional topological "objects" of various types, and to classify several classes of such objects using such…
Measuring similarity between complex objects is a fundamental task in many scientific fields. When objects are represented as graphs, graph similarity/distance measures offer a powerful framework for quantifying structural resemblance.…
In Graph Theory a number of results were devoted to studying the computational complexity of the number modulo 2 of a graph's edge set decompositions of various kinds, first of all including its Hamiltonian decompositions, as well as the…
Motivated by the problem about HOMO-LUMO separation that arises in mathematical chemistry, Fowler and Pisanski introduced the notion of the HL-index which measures how large in absolute value may be the median eigenvalues of a graph. In…
The Sombor index of a graph $G$ was recently introduced by Gutman from the geometric point of view, defined as $SO(G)=\sum_{uv\in E(G)}\sqrt{d(u)^2+d(v)^2}$, where $d(u)$ is the degree of a vertex $u$. For two real numbers $\alpha$ and…
We consider polynomial maps $f:\C\to\C$ of degree $d\ge 2$, or more generally polynomial maps from a finite union of copies of $\C$ to itself which have degree two or more on each copy. In any space $\p^{S}$ of suitably normalized maps of…
We give a method of generating strongly polynomial sequences of graphs, i.e., sequences $(H_{\mathbf{k}})$ indexed by a multivariate parameter $\mathbf{k}=(k_1,\ldots, k_h)$ such that, for each fixed graph $G$, there is a multivariate…
The automatic analysis of chemical literature has immense potential to accelerate the discovery of new materials and drugs. Much of the critical information in patent documents and scientific articles is contained in figures, depicting the…
In this paper, we investigate the degree of $h$-polynomials of edge ideals of finite simple graphs. In particular, we provide combinatorial formulas for the degree of the $h$-polynomial for various fundamental classes of graphs such as…
In this paper, we study the Steiner hyper-Wiener index of a graph, which is obtained from the standard hyper-Wiener index by replacing the classical graph distance with the Steiner distance. It is shown how this index is related to the…
The goal of this paper is to exhibit and analyze an algorithm that takes a given closed orientable hyperbolic surface and outputs an explicit Dirichlet domain. The input is a fundamental polygon with side pairings. While grounded in…
We derive formulas for the cd-index and the toric h-vector of a convex polytope P from a sweeping by a hyperplane. These arise from interpreting the corresponding S-shelling of the dual of P. We describe a partition of the faces of the…
Graph polynomials encode fundamental combinatorial invariants of graphs. Their computation is investigated using tree and path decomposition frameworks, with formal definitions of treewidth, k-trees, and pathwidth establishing the…
In this paper we prove Morse index theorems for a big class of constrained variational problems on graphs. Such theorems are useful in various physical and geometric applications. Our formulas compute the difference of Morse indices of two…
Hypergraphs are important objects to model ternary or higher-order relations of objects, and have a number of applications in analysing many complex datasets occurring in practice. In this work we study a new heat diffusion process in…
Let $G$ be a finite simple graph on $n$ vertices and $J_G$ denote the corresponding binomial edge ideal in the polynomial ring $S = K[x_1, \ldots, x_n, y_1, \ldots, y_n].$ In this article, we compute the Hilbert series of binomial edge…
We introduce a general class of combinatorial objects, which we call \emph{multi-complexes}, which simultaneously generalizes graphs, multigraphs, hypergraphs and simplicial and delta complexes. We introduce a natural algebra of…