Related papers: Degrees of interior polynomials and parking functi…

Erd\H{o}s, Fajtlowicz and Staton asked for the least integer $f(k)$ such that every graph with more than $f(k)$ vertices has an induced regular subgraph with at least $k$ vertices. Here we consider the following relaxed notions. Let $g(k)$…

A depth-first search version of Dhar's burning algorithm is used to give a bijection between the parking functions of a graph and labeled spanning trees, relating the degree of the parking function with the number of inversions of the…

This work builds on the notion of record of rooted trees. We provide an alternative definition of parking functions, derive from it a record-preserving bijection between rooted trees and parking functions, and establish a join…

In arbitrary Carnot groups we study intrinsic graphs of maps with horizontal target. These graphs are $C^1_H$ regular exactly when the map is uniformly intrinsically differentiable. Our first main result characterizes the uniformly…

The k-th power D^k of a directed graph D is defined to be the directed graph on the vertices of D with an arc from a to b in D^k iff one can get from a to b in D with exactly k steps. This notion is equivalent to the k-fold composition of…

For certain polynomials we relate the number of roots inside the unit circle with the index of a non-degenerate isolated umbilic point on a real analytic surface in Euclidean 3-space. In particular, for $N>0$ we prove that for a certain…

Let $p$ be a prime, let $1 \le t < d < p$ be integers, and let $S$ be a non-empty subset of $\mathbb{F}_p$. We establish that if a polynomial $P:\mathbb{F}_p^n \to \mathbb{F}_p$ with degree $d$ is such that the image $P(S^n)$ does not…

If ${\cal F}$ is a set of subgraphs $F$ of a finite graph $E$ we define a graph-counting polynomial $$ p_{\cal F}(z)=\sum_{F\in{\cal F}}z^{|F|} $$ In the present note we consider oriented graphs and discuss some cases where ${\cal F}$…

The local tree-width of a graph G=(V,E) is the function ltw^G: N -> N that associates with every natural number r the maximal tree-width of an r-neighborhood in G. Our main graph theoretic result is a decomposition theorem for graphs with…

We prove a formula for the multidegrees of a rational map defined by generalized monomials on a projective variety, in terms of integrals over an associated Newton region. This formula leads to an expression of the multidegrees as volumes…

Derivative polynomials in two variables are defined by repeated differentiation of the tangent and secant functions. We establish the connections between the coefficients of these derivative polynomials and the numbers of interior and left…

Let $G=(V,E)$ be a finite simple graph. In this paper, we study the degree of the $h$-polynomial of the edge ideal of $G$ in relation to the independence number of $G$. Our approach is based on the value of the independence polynomial of…

We define the branching ratio of the input tree of a node in a finite directed multigraph, prove that it exists for every node, and show that it is equal to the largest eigenvalue of the adjacency matrix of the induced subgraph determined…

We define a total order, which we call rooted order, on minimal generating set of $J(P_n)^s$ where $J(P_n)$ is the cover ideal of a path graph on $n$ vertices. We show that each power of a cover ideal of a path has linear quotients with…

The aim of this work is to extend to a general $S_m\times S_n$-module context the Grossman-Bizley paradigm that allows the enumeration of Dyck paths in a $m\times n$-rectangle. We obtain an explicit formula for the the "bi-Frobenius"…

Although Extension Perfect Roman Domination is NP-complete, all minimal (with respect to the pointwise order) perfect Roman dominating functions can be enumerated with polynomial delay. This algorithm uses a bijection between minimal…

The goal of this article is to obtain bounds on the coefficients of modular and integral flow and tension polynomials of graphs. To this end we make use of the fact that these polynomials can be realized as Ehrhart polynomials of inside-out…

We introduce a new one-variable polynomial invariant of graphs, which we call the skew characteristic polynomial. For an oriented simple graph, this is just the characteristic polynomial of its anti-symmetric adjacency matrix. For…

A theorem is proved concerning approximation of analytic functions by multivariate polynomials in the $s$-dimensional hypercube. The geometric convergence rate is determined not by the usual notion of degree of a multivariate polynomial,…

In earlier work we introduced the graph bracket polynomial of graphs with marked vertices, motivated by the fact that the Kauffman bracket of a link diagram D is determined by a looped, marked version of the interlacement graph associated…