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Blocking Sets of Index Three
Redei Blocking Sets Projective Triangle Projective Triad Combinatorics
2012/6/15
In this note we will provide proofs for the various statements that have been made in the literature about blocking sets of index three. Our aim is to clarify what is known about the characterization ...
Polymetric brick wall patterns and two-dimensional substitutions
Pseudo-self-similar tilings 2D-substitutions automatic sequences
2012/6/12
Polymetric walls are walls built from bricks in more than one size. Architects and builders want to built polymetric walls that satisfy certain structural and aesthetical constraints. In a recent pape...
A multivariate hook formula for labelled trees
hook formula tree enumeration representation theory of symmetric groups finite difference operators multivariate Lagrange inversion
2012/5/24
Several hook summation formulae for binary trees have appeared recently in the literature. In this paper we present an analogous formula for unordered increasing trees of size r, which involves r para...
Amenability, locally finite spaces, and bi-lipschitz embeddings
amenability isoperimetric constant embeddings into Hilbert spaces
2012/5/24
We define the isoperimetric constant for any locally finite metric space and we study the property of having isoperimetric constant equal to zero. This property, called Small Neighborhood property, cl...
Absolutely symmetric trees and complexity of natural number
Absolutely symmetric trees complexity of natural number Combinatorics
2012/5/24
We consider the rooted trees which not have isomorphic representation and introduce a conception of complexity a natural number also. The connection between quantity such trees with $n$ edges and a co...
A set $W\subseteq V(G)$ is called a resolving set, if for each two distinct vertices $u,v\in V(G)$ there exists $w\in W$ such that $d(u,w)\neq d(v,w)$, where $d(x,y)$ is the distance between the verti...
Branden's Conjectures on the Boros-Moll Polynomials
Boros-Moll Polynomials Real-rootedness Sturm sequence, 3-log-concavity
2012/5/24
We prove two conjectures of Br\"{a}nd\'{e}n on the real-rootedness of polynomials $Q_n(x)$ and $R_n(x)$ which are related to the Boros-Moll polynomials $P_n(x)$. In fact, we show that both $Q_n(x)$ an...
A quasi-tree expansion of the Krushkal polynomial
quasi-tree expansion Krushkal polynomial Combinatorics
2012/5/24
We introduce a generalization of the Krushkal polynomial to nonorientable surfaces, and prove that this polynomial has a natural quasi-tree expansion. This generalized Krushkal polynomial contains the...
Recently, it has been shown that a connected graph $\Gamma$ with $d+1$ distinct eigenvalues and odd-girth $2d+1$ is distance-regular. The proof of this result was based on the spectral excess theorem....
Areas of triangles and Beck's theorem in planes over finite fields
Areas of triangles Beck's theorem planes over finite fields Combinatorics
2012/5/9
It is shown that any subset $E$ of a plane over a finite field $\F_q$, of cardinality $|E|>q$ determines not less than $\frac{q-1}{2}$ distinct areas of triangles, moreover once can find such triangle...
The p-Domination Number of Complete Multipartite Graphs
p-domination set p-domination number complete multipartite graph
2012/5/9
Let $G=(V,E)$ be a graph and $p$ a positive integer. A subset $S\subseteq V$ is called a $p$-dominating set of $G$ if every vertex not in $S$ has at least $p$ neighbors in $S$. The $p$-domination numb...
Odd-order Cayley graphs with commutator subgroup of order pq are hamiltonian
Cayley graph hamiltonian cycle commutator subgroup
2012/5/9
We show that if G is a nontrivial, finite group of odd order, whose commutator subgroup [G,G] is cyclic of order p^m q^n, where p and q are prime, then every connected Cayley graph on G has a hamilton...
Lexicographic Shellability of Partial Involutions
Lexicographic Shellability of Partial Involutions Combinatorics Algebraic Geometry
2012/5/9
In this manuscript we study inclusion posets of Borel orbit closures on (symmetric) matrices. In particular, we show that the Bruhat poset of partial involutions is a lexicographiically shellable pose...
C-sortable words as green mutation sequences
Coxeter group quiver mutation cluster theory exchange graph
2012/5/9
Let Q be an acyclic quiver and s be a sequence s with elements in the vertex set Q_0. We describe a sequence of simple (backward) tilting in the bounded derived category D(Q), starting from the standa...
A connection between the bipartite complements of line graphs and the line graphs with two positive eigenvalues
line graphs graph spectra complements Courant-Weyl inequalities
2012/4/16
In 1974 Cvetkovi\'c and Simi\'c showed which graphs $G$ are the bipartite complements of line graphs. In 2002 Borovi\'canin showed which line graphs $L(H)$ have third largest eigenvalue $\lambda_3\leq...