Related papers: Integer diagonal forms for subset intersection rel…
A family of vectors $A \subset [k]^n$ is said to be intersecting if any two elements of $A$ agree on at least one coordinate. We prove, for fixed $k \ge 3$, that the size of a symmetric intersecting subfamily of $[k]^n$ is $o(k^n)$, which…
This paper considers the difficulty in the set-system approach to generalizing graph theory. These difficulties arise categorically as the category of set-system hypergraphs is shown not to be cartesian closed and lacks enough projective…
We introduce the \emph{Morse ensemble polynomial} $\ME_K(z_0,\ldots,z_d)$ of a finite simplicial complex $K$, defined as the generating function $\ME_K = \sum_M \prod_i z_i^{c_i(M)}$ over all acyclic matchings $M$ on the face poset of $K$,…
For each $n,r \geq 0$, let $KG(n,r)$ denote the Kneser Graph; that whose vertices are labeled by $r$-element subsets of $n$, and whose edges indicate that the corresponding subsets are disjoint. Fixing $r$ and allowing $n$ to vary, one…
Random graphs with latent geometric structure are popular models of social and biological networks, with applications ranging from network user profiling to circuit design. These graphs are also of purely theoretical interest within…
Let $G$ be a group. The intersection graph of cyclic subgroups of $G$, denoted by $\mathscr I_c(G)$, is a graph having all the proper cyclic subgroups of $G$ as its vertices and two distinct vertices in $\mathscr I_c(G)$ are adjacent if and…
Two families $\mathcal A$ and $\mathcal B$ of $k$-subsets of an $n$-set are called cross-intersecting if $A\cap B\ne\emptyset$ for all $A\in \mathcal A, B\in \mathcal B $. Strengthening the classical Erd\H os-Ko-Rado theorem, Pyber proved…
{\em Partial domination problem} is a generalization of the {\em minimum dominating set problem} on graphs. Here, instead of dominating all the nodes, one asks to dominate at least a fraction of the nodes of the given graph by choosing a…
Classical approaches often treat interaction as engineered product terms or as emergent patterns in flexible models, offering little control over how synergy or antagonism arises. We take a quantum-inspired view: following the Born rule…
We investigate the least common multiple of all subdeterminants, lcmd(A x B), of a Kronecker product of matrices, of which one is an integral matrix A with two columns and the other is the incidence matrix of a complete graph with n…
A collection of sets is {\em intersecting} if every two members have nonempty intersection. We describe the structure of intersecting families of $r$-sets of an $n$-set whose size is quite a bit smaller than the maximum ${n-1 \choose r-1}$…
Determinantal ideals of graphs generalize, among others, the spectrum and the Smith normal form (SNF) of integer matrices associated to graphs. In this work we investigate the relationship of the spectrum and the SNF with the determinantal…
Intersection graphs are very important in both theoretical as well as application point of view. Depending on the geometrical representation, different type of intersection graphs are defined. Among them interval, circular-arc, permutation,…
A string graph is an intersection graph of curves in the plane. A $k$-string graph is a graph with a string representation in which every pair of curves intersects in at most $k$ points. We introduce the class of $(=k)$-string graphs as a…
Let $V$ be a nonempty finite set and $A=(a_{ij})_{i,j\in V}$ be a matrix with entries in a field $\mathbb{K}$. For a subset $X$ of $V$, we denote by $A[X]$ the submatrix of $A$ having row and column indices in $X$. We study the following…
We call a family of sets intersecting, if any two sets in the family intersect. In this paper we investigate intersecting families $\mathcal{F}$ of $k$-element subsets of $[n]:=\{1,\ldots, n\},$ such that every element of $[n]$ lies in the…
A set of permutations $I \subset S_n$ is said to be {\em k-intersecting} if any two permutations in $I$ agree on at least $k$ points. We show that for any $k \in \mathbb{N}$, if $n$ is sufficiently large depending on $k$, then the largest…
Recently there is huge interest in graph theory and intensive study on computing integer powers of matrices. In this paper, we investigate relationships between one type of graph and well-known Fibonacci sequence. In this content, we…
In this paper, we demonstrate a useful interaction between the theory of clique partitions, edge clique covers of a graph, and the spectra of graphs. Using a clique partition and an edge clique cover of a graph we introduce the notion of a…
We explicitly characterize when the Milnor number at the origin of a polynomial or power series (over an algebraically closed field k of arbitrary characteristic) is the minimum of all polynomials with the same Newton diagram, which…