Related papers: Determinants of modular Collatz graphs and variant…
For classical knots, Murasugi showed that the determinant modulo $8$ is classified by the Arf invariant. Boden and Karimi introduced a determinant for checkerboard colorable virtual knots. We prove that this determinant modulo $8$ is…
Among all the dynamical modular curves associated to quadratic polynomial maps, we determine which curves have infinitely many quadratic points. This yields a classification statement on preperiodic points for quadratic polynomials over…
This note collects some results and conjectures for the generating functions of the Hankel determinants of certain polynomials which are related to Motzkin paths.
We compare two classes of functions arising from genus-one superstring amplitudes: modular and holomorphic graph functions. We focus on their analytic properties, we recall the known asymptotic behaviour of modular graph functions and we…
We present simple graph-theoretic characterizations of Cayley graphs for monoids, semigroups and groups. We extend these characterizations to commutative monoids, semilattices, and abelian groups.
We study the real components of modular curves. Our main result is an abstract group-theoretic description of the real components of a modular curve defined by a congruence subgroup of level N in terms of the corresponding subgroup of…
In this paper we study unimodality problems for the independence polynomial of a graph, including unimodality, log-concavity and reality of zeros. We establish recurrence relations and give factorizations of independence polynomials for…
We prove new determinantal identities for a family of flagged Schur polynomials. As a corollary of these identities we obtain determinantal expressions of Schubert polynomials for certain vexillary permutations.
This Note gives conditions that must be imposed to algebraic multilevel discretizations involving at the same time nodal and edge elements so that a gradient-prolongation commutativity condition will be satisfied; this condition is very…
We consider the relation between various permutation statistics and properties of permutation tableaux. We answer some of the questions of Steingrimsson and Williams (math.CO/0507149), in particular, on the distribution of the bistatistic…
We study a class of rotation invariant determinantal ensembles in the complex plane; examples include the eigenvalues of Gaussian random matrices and the roots of certain families of random polynomials. The main result is a criteria for a…
Using recurrence matrices, defined and described with some details, we study a few determinants related to evaluations of binomial coefficients on Dirichlet characters modulo 2, 4 and 8.
Topics concerning metric dimension related invariants in graphs are nowadays intensively studied. This compendium of combinatorial and computational results on this topic is an attempt of surveying those contributions that are of the…
This work presents a detailed analysis of the combinatorics of modular operads. These are operad-like structures that admit a contraction operation as well as an operadic multiplication. Their combinatorics are governed by graphs that admit…
This paper investigates some combinatorial and algebraic properties of a Witt type formula for graphs.
Modular operads are a special type of operad: in fact, they bear the same relationship to operads that graphs do to trees (i.e. simply connected graphs). One of the basic examples of a modular operad is the collection of…
We study correlation functions of the characteristic polynomials in coupled matrix models based on the Schur polynomial expansion, which manifests their determinantal structure.
We study modular ortholattices in the variety generated by the finite dimensional ones from an equational and geometric point of view. We relate this to coordinatization results.
We obtain a complete description of the planar cubic Cayley graphs, providing an explicit presentation and embedding for each of them. This turns out to be a rich class, comprising several infinite families. We obtain counterexamples to…
We give an efficient construction of a reasonably small dominating set in a circulant graph on $n$ notes and $k$ distinct chord lengths. This result is based on bounds on some double exponential sums. .