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A cornerstone of geometric reconstruction, rotation averaging seeks the set of absolute rotations that optimally explains a set of measured relative orientations between them. In addition to being an integral part of bundle adjustment and…

Motivated by some known problems concerning combinatorial structures associated with finite one-dimensional affine permutation groups, we study subgroups which are closed in $\operatorname{\Gamma{L}}_1(q)$. This brings us to a description…

F-index of a graph is the sum of the cube of the degrees of the vertices. In this paper, we investigate the F-indices of unicyclic graphs by introducing some transformation, and characterize the unicyclic graphs with the first five largest…

We present the first truly subcubic, combinatorial algorithm for detecting an induced $4$-cycle in a graph. The running time is $O(n^{2.84})$ on $n$-node graphs, thus separating the task of detecting induced $4$-cycles from detecting…

Consider an ideal $I\subset K[x_1,..., x_n]$, with $K$ an arbitrary field, generated by monomials of degree two. Assuming that $I$ does not have a linear resolution, we determine the step $s$ of the minimal graded free resolution of $I$…

We define a new combinatorial object, which we call a labeled hypergraph, uniquely associated to any square-free monomial ideal. We prove several upper bounds on the regularity of a square-free monomial ideal in terms of simple…

We revisit Min-Mean-Cycle, the classical problem of finding a cycle in a weighted directed graph with minimum mean weight. Despite an extensive algorithmic literature, previous work falls short of a near-linear runtime in the number of…

Universal cycles, such as De Bruijn cycles, are cyclic sequences of symbols that represent every combinatorial object from some family exactly once as a consecutive subsequence. Graph universal cycles are a graph analogue of universal…

A tight Hamilton cycle in a $k$-uniform hypergraph ($k$-graph) $G$ is a cyclic ordering of the vertices of $G$ such that every set of $k$ consecutive vertices in the ordering forms an edge. R\"{o}dl, Ruci\'{n}ski, and Szemer\'{e}di proved…

We study several ideal-based constructions in the context of singular stationarity. By combining methods of strong ideals, supercompact embeddings, and Prikry-type posets, we obtain three consistency results concerning mutually stationary…

We study the defining equations of the Rees algebra of square-free monomial ideals in a polynomial ring over a field. We determine that when an ideal $I$ is generated by $n$ square-free monomials of the same degree then $I$ has relation…

In this paper we study graded ideals I in a polynomial ring S such that the numerical function f(k)=depth(S/I^k) is constant. We show that, if (i) the Rees algebra of I is Cohen-Macaulay, (ii) the cohomological dimension of I is not larger…

The Feynman rules assign to every graph an integral which can be written as a function of a scaling parameter L. Assuming L for the process under consideration is very small, so that contributions to the renormalizaton group are small, we…

Let $G$ be a finite simple graph, and $J_G$ denote the binomial edge ideal of $G$. In this article, we first compute the $\mathrm{v}$-number of binomial edge ideals corresponding to Cohen-Macaulay closed graphs. As a consequence, we obtain…

We investigate the average hitting times of simple random walks on the $k$-th power graph $C_N^k$ of the cycle graph $C_N$. First, we show that the average hitting times are characterized by a difference equation corresponding to the graph…

We study the set ${\cal L}(G)$ of lengths of all cycles that appear in a random $d$-regular $G$ on $n$ vertices for a fixed $d\geq 3$, as well as in Erd\H{o}s--R\'enyi random graphs on $n$ vertices with a fixed average degree $c>1$.…

Consider factorizations into transpositions of an n-cycle in the symmetric group S_n. To every such factorization we assign a monomial in variables w_{ij} that retains the transpositions used, but forgets their order. Summing over all…

We introduce a monomial ideal whose standard monomials encode the vertices of all fibers of a lattice. We study the minimal generators, the radical, the associated primes and the primary decomposition of this ideal, as well as its relation…

Let $G$ be the circulant graph $C_n(S)$ with $S\subseteq\{ 1,\ldots,\left \lfloor\frac{n}{2}\right \rfloor\}$ and let $I(G)$ be its edge ideal in the ring $K[x_0,\ldots,x_{n-1}]$. Under the hypothesis that $n$ is prime we : 1) compute the…

For a polynomial ring S in n variables, we consider the natural action of the symmetric group S_n on S by permuting the variables. For an S_n-invariant monomial ideal I in S and j >= 0, we give an explicit recipe for computing the modules…