A cyclic
(
n
3
)
configuration is a combinatorial configuration whose automorphism group contains a cyclic permutation of the points of the configuration; that is, the points of the configuration may ...be considered to be elements of
Z
n
, and the lines of the configuration as cyclic shifts of a single fixed starting block 0,
a
,
b
, where
a
,
b
∈
Z
n
. We denote such configurations as
Cyc
n
(
0
,
a
,
b
)
. One of the fundamental questions in the study of configurations is that of
geometric realizability
. In the case where
n
=
2
m
, it is combinatorially possible to divide the points and lines of the configuration into two classes according to parity, so it is natural to ask whether the configuration can be realized using those classes. We provide methods for producing geometric realizations of configurations
Cyc
2
m
(
0
,
a
,
b
)
that have two symmetry classes under the maximal rotational subgroup of the geometric symmetry group (that is, chiral astral realizations), and we provide a number of constraints on
a
and
b
that guarantee such a realization exists. Experiments on up to 500 points suggest that, with the exception of some small sporadic examples and a single infinite family
Cyc
2
(
k
+
1
)
(
0
,
1
,
k
)
,
k
≥
3
and
k
odd, all cyclic
(
2
m
3
)
configurations are realizable as geometric chiral astral configurations using the methods described in this paper.
Circular bidiagonal pairs Terwilliger, Paul; Žitnik, Arjana
Linear algebra and its applications,
07/2023, Letnik:
669
Journal Article
Recenzirano
Odprti dostop
A square matrix is said to be circular bidiagonal whenever (i) each nonzero entry is on the diagonal, or the subdiagonal, or in the top-right corner; (ii) each subdiagonal entry is nonzero, and the ...entry in the top-right corner is nonzero. Let F denote a field, and let V denote a nonzero finite-dimensional vector space over F. We consider an ordered pair of F-linear maps A:V→V and A⁎:V→V that satisfy the following two conditions:•there exists a basis for V with respect to which the matrix representing A is circular bidiagonal and the matrix representing A⁎ is diagonal;•there exists a basis for V with respect to which the matrix representing A⁎ is circular bidiagonal and the matrix representing A is diagonal. We call such a pair a circular bidiagonal pair on V. We classify the circular bidiagonal pairs up to affine equivalence. There are two infinite families of solutions, which we describe in detail.
Let Γ denote a finite, undirected, connected graph, with vertex set X. Fix a vertex x∈X. Associated with x is a certain subalgebra T=T(x) of MatX(C), called the subconstituent algebra. The algebra T ...is semisimple. Hora and Obata introduced a certain subalgebra Q⊆T, called the quantum adjacency algebra. The algebra Q is semisimple. In this paper we investigate how Q and T are related. In many cases Q=T, but this is not true in general. To clarify this issue, we introduce the notion of quasi-isomorphic irreducible T-modules. We show that the following are equivalent: (i) Q≠T; (ii) there exists a pair of quasi-isomorphic irreducible T-modules that have different endpoints. To illustrate this result we consider two examples. The first example concerns the Hamming graphs. The second example concerns the bipartite dual polar graphs. We show that for the first example Q=T, and for the second example Q≠T.
Classification of Cubic Tricirculant Nut Graphs Damnjanović, Ivan; Bašić, Nino; Pisanski, Tomaž ...
The Electronic journal of combinatorics,
05/2024, Letnik:
31, Številka:
2
Journal Article
Recenzirano
Odprti dostop
A nut graph is a simple graph whose adjacency matrix has the eigenvalue zero with multiplicity one such that its corresponding eigenvector has no zero entries. It is known that there exist no cubic ...circulant nut graphs. A bicirculant (resp. tricirculant) graph is defined as a graph that admits a cyclic group of automorphisms having two (resp. three) orbits of vertices of equal size. We show that there exist no cubic bicirculant nut graphs and we provide a full classification of cubic tricirculant nut graphs.
Let C denote the field of complex numbers, and fix a nonzero q∈C such that q4≠1. Define a C-algebra Δq by generators and relations in the following way. The generators are A, B, C. The relations ...assert that each ofA+qBC−q−1CBq2−q−2,B+qCA−q−1ACq2−q−2,C+qAB−q−1BAq2−q−2 is central in Δq. The algebra Δq is called the universal Askey–Wilson algebra. Let Γ denote a distance-regular graph that has q-Racah type. Fix a vertex x of Γ and let T=T(x) denote the corresponding subconstituent algebra. In this paper we discuss a relationship between Δq and T. Assuming that every irreducible T-module is thin, we display a surjective C-algebra homomorphism Δq→T. This gives a Δq action on the standard module of T.
A half-arc-transitive graph is a regular graph that is both vertex- and edge-transitive, but is not arc-transitive. If such a graph has finite valency, then its valency is even, and greater than 2. ...In 1970, Bouwer proved that there exists a half-arc-transitive graph of every even valency greater than 2, by giving a construction for a family of graphs now known as B(k,m,n), defined for every triple (k,m,n) of integers greater than 1 with 2m≡1modn. In each case, B(k,m,n) is a 2k-valent vertex- and edge-transitive graph of order mnk−1, and Bouwer showed that B(k,6,9) is half-arc-transitive for all k>1.
For almost 45 years the question of exactly which of Bouwer’s graphs are half-arc-transitive and which are arc-transitive has remained open, despite many attempts to answer it. In this paper, we use a cycle-counting argument to prove that almost all of the graphs constructed by Bouwer are half-arc-transitive. In fact, we prove that B(k,m,n) is arc-transitive only when n=3, or (k,n)=(2,5), or (k,m,n)=(2,3,7) or (2,6,7) or (2,6,21). In particular, B(k,m,n) is half-arc-transitive whenever m>6 and n>5. This gives an easy way to prove that there are infinitely many half-arc-transitive graphs of each even valency 2k>2.
Datamatrix code is a type of 2D codes that can encode much more data on the same or smaller area than the linear barcodes. This makes 2D codes usable for marking even very small items. 2D codes can ...be decoded by the readers in retails but also with the mobile phones equipped with camera and appropriate software. 2D codes can be depicted in different materials or printed on different printing substrates. The application area of the codes is broad, from magazines and newspapers to posters and packaging. Successful reading of 2D codes is possible if the code is printed in appropriate contrast between the printing ink and substrate, like black ink printed on white matt paper. Problems can occur if the code is printed in colors. The readability of 2D Datamatrix codes printed in cyan, magenta, yellow and black was studied. Yellow is proved to be poorly readable. In addition, the bi-colored and multi-colored 2D Datamatrix codes were studied. When four colors are used in creation of the 2D Datamatrix code, poorly readable elements, yellow codewords, may cause the reading failure. 2D Datamatix codes are capable to ensure good readability even if they contain a defined number of poorly readable codewords due to the Reed – Solomon error-correction system. The aim of the study was to investigate the effect of using yellow printed, poorly readable, codewords in the multi-colored 2D Datamatrix code on the code readability.
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