Related papers: Fullerenes with the maximum Clar number

For a given natural number $n$, the second part of Hilbert's 16th Problem asks whether there exists a finite upper bound for the maximum number of limit cycles that planar polynomial vector fields of degree $n$ can have. This maximum number…

Consider a face-to-face parallelohedral tiling of $\mathbb R^d$ and a $(d-k)$-dimensional face $F$ of the tiling. We prove that the valence of $F$ (i.e. the number of tiles containing $F$ as a face) is not greater than $2^k$. If the tiling…

We prove that $\det A\leq 6^\frac{n}{6}$ whenever $A\in\{0,1\}^{n\times n}$ contains at most $2n$ ones. We also prove an upper bound on the determinant of matrices with the $k$-consecutive ones property, a generalisation of the consecutive…

A graph is called (generically) rigid in R^d if, for any choice of sufficiently generic edge lengths, it can be embedded in R^d in a finite number of distinct ways, modulo rigid transformations. Here, we deal with the problem of determining…

A fullerene graph can be embedded in a piecewise linear 2-manifold with each non-hexagonal carbon ring corresponding to a cone vertex. Adjacent two or three such vertices can be combined as a cluster cut out from a parent cone round a…

This paper investigates the maximum spectral radius of planar graphs with concrete fixed number of vertices, providing some tight bounds on the maximum spectral radius of general planar graph resorting to its order, and confirming that…

The celebrated upper bound theorem of McMullen determines the maximal number of extreme points of a polyhedron in terms of its dimension and the number of constraints which define it, showing that the maximum is attained by the polar of the…

We determine all Chern numbers of smooth complex projective varieties of dimension at least four which are determined up to finite ambiguity by the underlying smooth manifold. We also give an upper bound on the dimension of the space of…

We show that for $n$ at least $10^{11}$, any 2-coloring of the $n$-dimensional grid $[4]^n$ contains a monochromatic combinatorial line. This is a special case of the Hales-Jewett Theorem, to which the best known general upper bound is due…

Let $\mathrm{rex}(n, F)$ denote the maximum number of edges in an $n$-vertex graph that is regular and does not contain $F$ as a subgraph. We give lower bounds on $\mathrm{rex}(n, F)$, that are best possible up to a constant factor, when…

We consider the following question, motivated by the enumeration of fullerenes. A fullerene patch is a 2-connected plane graph G in which inner faces have length 5 or 6, non-boundary vertices have degree 3, and boundary vertices have degree…

We investigate terminal-pairability properties of complete graphs and improve the known bounds in two open problems. We prove that the complete graph $K_n$ on $n$ vertices is terminal-pairable if the maximum degree $\Delta$ of the…

We prove the existence of curves of genus $7$ and $12$ over the field with $11^5$ elements, reaching the Hasse-Weil-Serre upper bound. These curves are quotients of modular curves and we give explicit equations. We compute the number of…

The Clar covering polynomial (also called Zhang-Zhang polynomial in some chemical literature) of a hexagonal system is a counting polynomial for some types of resonant structures called Clar covers, which can be used to determine Kekul\'e…

The planar Turan number $\textrm{ex}_{\mathcal{P}}(C_{\ell},n)$ is the largest number of edges in an $n$-vertex planar graph with no $\ell$-cycle. For $\ell\in \{3,4,5,6\}$, upper bounds on $\textrm{ex}_{\mathcal{P}}(C_{\ell},n)$ are known…

We introduce a new approach and prove that the maximum number of triangles in a $C_5$-free graph on $n$ vertices is at most $$(1 + o(1)) \frac{1}{3 \sqrt 2} n^{3/2}.$$ We also show a connection to $r$-uniform hypergraphs without (Berge)…

For each $d>0$, we find all the smallest fullerenes for which the least distance between two pentagons is $d$. We also show that for each $d$ there is an $h_d$ such that fullerenes with pentagons at least distance $d$ apart and any number…

We show that a graph with $n$ vertices and vertex cover of size $k$ has at most $4^k + n$ potential maximal cliques. We also show that for each positive integer $k$, there exists a graph with vertex cover of size $k$, $O(k^2)$ vertices, and…

We prove an upper bound of the form $2^{O(d^2 \mathrm{polylog}\,d)}$ on the number of affine (resp. linear) equivalence classes of, by increasing order of generality, 2-level d-polytopes, d-cones and d-configurations. This in particular…

We show that the number of all maximal $\alpha$-gapped repeats and palindromes of a word of length $n$ is at most $3(\pi^2/6 + 5/2) \alpha n$ and $7 (\pi^2 / 6 + 1/2) \alpha n - 5 n - 1$, respectively.