Related papers: A generalized Beraha conjecture for non-planar gra…

The generalized theta graph \Theta_{s_1,...,s_k} consists of a pair of endvertices joined by k internally disjoint paths of lengths s_1,...,s_k \ge 1. We prove that the roots of the chromatic polynomial $pi(\Theta_{s_1,...,s_k},z) of a…

We prove that if $p\ge n^{-\frac{1}{3}+\beta}$ for some $\beta > 0$, then asymptotically almost surely the binomial random graph $G(n,p)$ has a $K_3$-packing containing all but at most $n + O(1)$ edges. Similarly, we prove that if $d \ge…

Let $\mathfrak q$ be a finite-dimensional Lie algebra, $\vartheta\in Aut(\mathfrak q)$ a finite order automorphism, and $\mathfrak q_0$ the subalgebra of fixed points of $\vartheta$. Using $\vartheta$ one can construct a pencil $\mathcal P$…

We say that a plane set $A$ is {\it graph-null,} if there is a function $g\colon [0,1] \to \mathbb{R}$ such that $\lambda_2 (A+{\rm graph}\, g)=0$. A plane set $A$ has the {\it translational Kakeya property} if, for every translated copy…

We prove that if $G$ is a graph and $r_1, ..., r_k \in \mathbb{Z}_{\geq 0}$ such that $\sum_{i=1}^k r_i \geq \Delta(G) + 2 - k$ then $V(G)$ can be partitioned into sets $V_1, ..., V_k$ such that $\Delta(G[V_i]) \leq r_i$ and $G[V_i]$…

We propose the Kazakov-Migdal model on graphs and show that, when the parameters of this model are appropriately tuned, the partition function is represented by the unitary matrix integral of an extended Ihara zeta function, which has a…

Graphical parking functions, or $G$-parking functions, are a generalization of classical parking functions which depend on a connected multigraph $G$ having a distinguished root vertex. Gaydarov and Hopkins characterized the relationship…

For a simple connected graph $G$, the $Q$-generating function of the numbers $N_k$ of semi-edge walks of length $k$ in $G$ is defined by $W_Q(t)=\sum\nolimits_{k = 0}^\infty {N_k t^k }$. This paper reveals that the $Q$-generating function…

We study properties of the Potts model partition function $Z(H_m,q,v)$ on $m$'th iterates of Hanoi graphs, $H_m$, and use the results to draw inferences about the $m \to \infty$ limit that yields a self-similar Hanoi fractal, $H_\infty$. We…

We continue the study of the $k$-cut complex $\Delta_k(G)$ of a graph $G$ initiated in the paper of Bayer, Denker, Jeli\'c Milutinovi\'c, Rowlands, Sundaram and Xue [Topology of cut complexes of graphs, SIAM J. on Discrete Math. 38(2):…

In this paper, we study the generalized gapped k-mer filters and derive a closed form solution for their coefficients. We consider nonnegative integers $\ell$ and $k$, with $k\leq \ell$, and an $\ell$-tuple $B=(b_1,\ldots,b_{\ell})$ of…

We find zero-free regions in the complex plane at large |q| for the multivariate Tutte polynomial (also known in statistical mechanics as the Potts-model partition function) Z_G(q,w) of a graph G with general complex edge weights w = {w_e}.…

The Kneser graph $KG_{n,k}$ is the graph whose vertices are the $k$-element subsets of $[n],$ with two vertices adjacent if and only if the corresponding sets are disjoint. A famous result due to Lov\'asz states that the chromatic number of…

We study the properties of a function $\psi(z, q)$ (the generalized polygamma function), intimately connected with the Hurwitz zeta function and defined for complex values of the variables $z$ and $q$, which is entire in the variable $z$…

We perform extensive Monte Carlo simulations of the 10-state Potts model on quenched two-dimensional $\Phi^3$ gravity graphs to study the effect of quenched coordination number randomness on the nature of the phase transition, which is…

This paper lays out the foundations of graded $K$-theory for Leavitt algebras associated with higher-rank graphs, also known as Kumjian-Pask algebras, establishing it as a potential tool for their classification. For a row-finite $k$-graph…

In the paper, we introduce $q$-deformations of the Riemann zeta function, extend them to the whole complex plane, and establish certain estimates of the number of roots. The construction is based on the recent difference generalization of…

We consider the problem of constructing an abstract $(n+1)$-polytope $Q$ with $k$ facets isomorphic to a given $n$-polytope $P$, where $k \geq 3$. In particular, we consider the case where we want $Q$ to be $(n-2,n)$-flat, meaning that…

We propose a novel fermionic model on the graphs. The Dirac operator of the model consists of deformed incidence matrices on the graph and the partition function is given by the inverse of the graph zeta function. We find that the…

We first show the existence and nature of convergence to a limiting set of roots for polynomials in a three-term recurrence of the form $p_{n+1}(z) = Q_k(z)p_{n}(z)+ \gamma p_{n-1}(z)$ as $n$ $\rightarrow$ $\infty$, where the coefficient…