Related papers: A Conjectural Inequality for Visible Points in Lat…

Suppose that $[n]=\left\{0,1,2,...,n\right\}$ is a set of non-negative integers and $h,k \in [n]$. The $L(h,k)$-labeling of graph $G$ is the function $l:V(G)\rightarrow[n]$ such that $\left|l(u)-l(v)\right|\geq h$ if the distance $d(u,v)$…

We prove two conjectures in spectral extremal graph theory involving the linear combinations of graph eigenvalues. Let $\lambda_1(G)$ be the largest eigenvalue of the adjacency matrix of a graph $G$, and $\bar{G}$ be the complement of $G$.…

Let S=(s_1,s_2,..., s_m) and T = (t_1,t_2,..., t_n) be vectors of non-negative integers with sum_{i=1}^{m} s_i = sum_{j=1}^n t_j. Let B(S,T) be the number of m*n matrices over {0,1} with j-th row sum equal to s_j for 1 <= j <= m and k-th…

For a planar integral lattice $L$, let $\nu(L)$ denote the square-free part of the integer $D(L)^2$, where $D(L)$ stands for the area of a fundamental parallelogram of $L$. For each odd integer $n$ with $3 \leq n<29$, a planar lattice $L$…

Visibility problems have been investigated for a long time under different assumptions as they pose challenging combinatorial problems and are connected to robot navigation problems. The mutual-visibility problem in a graph $G$ of $n$…

We have developed a parallel algorithm that allows us to enumerate the number of self-avoiding polygons on the square lattice to perimeter length 110. We have also extended the series for the first 10 area-weighted moments and the radius of…

The concept of mutual visibility in a graph encodes combinatorial information about vertex subsets with prescribed visibility properties and serves as a useful algebraic invariant. In this paper, we derive algebraic conditions for the…

Let $L$ be a slim, planar, semimodular lattice (slim means that it does not contain an ${\mathsf M}_3$-sublattice). We call the interval $I = [o, i]$ of $L$ \emph{rectangular}, if there are complementary $a, b \in I$ such that $a$ is to the…

We attach to each $\langle 0, \vee \rangle$-semilattice a graph $\boldsymbol{G}_{\boldsymbol{S}}$ whose vertices are join-irreducible elements of $\boldsymbol{S}$ and whose edges correspond to the reflexive dependency relation. We study…

We consider undirected simple finite graphs. The sets of vertices and edges of a graph $G$ are denoted by $V(G)$ and $E(G)$, respectively. For a graph $G$, we denote by $\delta(G)$ and $\eta(G)$ the least degree of a vertex of $G$ and the…

Let $d\geq 3$ be a fixed integer and $A$ be the adjacency matrix of a random $d$-regular directed or undirected graph on $n$ vertices. We show there exist constants $\mathfrak d>0$, \begin{align*} {\mathbb P}(\text{$A$ is singular in…

For $S \subseteq \mathbb{R}$, positive integer $n$, and $d > 0$, let $G(S^n, d)$ be the graph whose vertex set is $S^n$ where any two vertices are adjacent if and only if they are Euclidean distance $d$ apart. The primary question we will…

Consider the sequence $\mathcal{V}(2,n)$ constructed in a greedy fashion by setting $a_1 = 2$, $a_2 = n$ and defining $a_{m+1}$ as the smallest integer larger than $a_m$ that can be written as the sum of two (not necessarily distinct)…

We experimentally demonstrate PT-symmetric optical lattices with periodical gain and loss profiles in a coherently-prepared four-level N-type atomic system. By appropriately tuning the pertinent atomic parameters, the onset of PT-symmetry…

It is conjectured since long that for any convex body $P\subset \mathbb{R}^n$ there exists a point in its interior which belongs to at least $2n$ normals from different points on the boundary of $P$. The conjecture is known to be true for…

For any $n > 0$ and $0 \leq m < n$, let $P_{n,m}$ be the poset of projective equivalence classes of $\{-,0,+\}$-vectors of length $n$ with sign variation bounded by $m$, ordered by reverse inclusion of the positions of zeros. Let…

In a coupled network cells can interact in several ways. There is a vast literature from the last twenty years that investigates this interacting dynamics under a graph theory formalism, namely as a graph endowed with an input-equivalence…

Let PR$[n]$ be the graph whose vertices are $2,3,\ldots,n$ with vertex $v$ adjacent to vertex $w$ if and only if $\gcd(v,w)>1$. It is shown that $\pi(n)$, the the number of primes no more than $n$, equals the Lov\'{a}sz number of this…

We show that the isotropic discrepancy of a lattice point set can be bounded from below and from above in terms of the spectral test of the corresponding integration lattice. From this we deduce that the isotropic discrepancy of any…

For fixed coprime polynomials $U,V \in \mathbb{F}_q [T]$ with degrees of different parities, and a general polynomial $A \in \mathbb{F}_q [T]$, define the arithmetic function $S_{U,V} (A)$ to be the number of representations of $A$ of the…