Electronic Notes in Discrete Mathematics最新文献

This presentation is an exposition of an application of the theory of recurrence relations to enumerating strings over an alphabet with a forbidden factor (consecutive substring). As an illustration we examine the case of binary strings with a forbidden factor of k consecutive symbols 1 for given k, using generating function techniques that deserve to be better known.

This allows us to derive a known upper bound for the number of prefix normal binary words: words with the property that no factor has more occurrences of the symbol 1 than the prefix of the same length. Such words arise in the context of indexed binary jumbled pattern matching.

We describe work on the relationship between the independently-studied polygon-circle graphs and word-representable graphs.

A graph G = (V, E) is word-representable if there exists a word w over the alpha-bet V such that letters x and y form a subword of the form xyxy ⋯ or yxyx ⋯ iff xy is an edge in E. Word-representable graphs generalise several well-known and well-studied classes of graphs [S. Kitaev, A Comprehensive Introduction to the Theory of Word-Representable Graphs, Lecture Notes in Computer Science 10396 (2017) 36–67; S. Kitaev, V. Lozin, “Words and Graphs”, Springer, 2015]. It is known that any word-representable graph is k-word-representable, that is, can be represented by a word having exactly k copies of each letter for some k dependent on the graph. Recognising whether a graph is word-representable is NP-complete ([S. Kitaev, V. Lozin, “Words and Graphs”, Springer, 2015, Theorem 4.2.15]). A polygon-circle graph (also known as a spider graph) is the intersection graph of a set of polygons inscribed in a circle [M. Koebe, On a new class of intersection graphs, Ann. Discrete Math. (1992) 141–143]. That is, two vertices of a graph are adjacent if their respective polygons have a non-empty intersection, and the set of polygons that correspond to vertices in this way are said to represent the graph. Recognising whether an input graph is a polygon-circle graph is NP-complete [M. Pergel, Recognition of polygon-circle graphs and graphs of interval filaments is NP-complete, Graph-Theoretic Concepts in Computer Science: 33rd Int. Workshop, Lecture Notes in Computer Science, 4769 (2007) 238–247]. We show that neither of these two classes is included in the other one by showing that the word-representable Petersen graph and crown graphs are not polygon-circle, while the non-word-representable wheel graph W₅ is polygon-circle. We also provide a more refined result showing that for any k ≥ 3, there are k-word-representable graphs which are neither (k −1)-word-representable nor polygon-circle.

For a lattice, finding a nonzero shortest vector is computationally difficult in general. The problem becomes quite complicated even when the dimension of the lattice is five. There are two related notions of reduced bases, say, Minkowski-reduced basis and greedy-reduced basis. When the dimension becomes d = 5, there are greedy-reduced bases without achieving the first minimum while any Minkowski-reduced basis contains the shortest four linearly independent vectors. This suggests that the notion of Minkowski-reduced basis is somewhat strong and the notion of greedy-reduced basis is too weak for a basis to achieve the first minimum of the lattice. In this work, we investigate a more appropriate condition for a basis to achieve the first minimum for d = 5. We present a minimal sufficient condition, APG⁺, for a five dimensional lattice basis to achieve the first minimum in the sense that any proper subset of the required inequalities is not sufficient to achieve the first minimum.

This is a generalisation of a chessboard pebbling problem (and the associated escapability-result) from one quadrant of an infinite integer lattice (as discussed by Chung et. al. in 1995 [Chung, F., Graham, R., Morrison, J. and Odlyzko, A., Pebbling a Chessboard, The American Mathematical Monthly, 102(2), 1995, p. 113.]) to four quadrants.

The $Z_{2^s}$-additive codes are subgroups of $Z_{2^s}^n$, and can be seen as a generalization of linear codes over $Z_2$ and $Z_4$. A $Z_{2^s}$-linear Hadamard code is a binary Hadamard code which is the Gray map image of a $Z_{2^s}$-additive code. It is known that either the rank or the dimension of the kernel can be used to give a complete classification for the $Z_4$-linear Hadamard codes. However, when $s>2$, the dimension of the kernel of $Z_{2^s}$-linear Hadamard codes of length $2^t$ only provides a complete classification for some values of t and s. In this paper, the rank of these codes is given for $s=3$. Moreover, it is shown that this invariant, along with the dimension of the kernel, provides a complete classification, once $t\ge 3$ is fixed. In this case, the number of nonequivalent such codes is also established.

The cut polytope CUT(n) is the convex hull of the cut vectors in a complete graph with vertex set {1,…, n}. It is well known in the area of combinatorial optimization and recently has also been studied in a direct relation with admissible correlations of symmetric Bernoulli random variables. That probabilistic interpretation is a starting point of this work in conjunction with a natural binary encoding of the CUT(n). We show that for any n, with appropriate scaling, all encoded vertices of the polytope 1-CUT(n) are approximately on the line $y=x-1/2$.

This note focuses on the problem of generalized exponential stability for linear time-varying discrete-time systems in Banach spaces. Characterizations for this concept are presented and connections with the classical concept of exponential stability existent in the literature is pointed out. An illustrative example clarifies the implications between these concepts.

In graphs, the concept of adjacency is clearly defined: it is a pairwise relationship between vertices. Adjacency in hypergraphs has to integrate hyperedge multi-adicity: the concept of adjacency needs to be defined properly by introducing two new concepts: k-adjacency – k vertices are in the same hyperedge – and e-adjacency – vertices of a given hyperedge are e-adjacent. In order to build a new e-adjacency tensor that is interpretable in terms of hypergraph uniformisation, we designed two processes: the first is a hypergraph uniformisation process (HUP) and the second is a polynomial homogeneisation process (PHP). The PHP allows the construction of the e-adjacency tensor while the HUP ensures that the PHP keeps interpretability. This tensor is symmetric and can be fully described by the number of hyperedges; its order is the range of the hypergraph, while extra dimensions allow to capture additional hypergraph structural information including the maximum level of k-adjacency of each hyperedge. Some results on spectral analysis are discussed.

Integer sequences express and capture many important concepts in number theory. They also arise naturally in some parts of dynamical systems, and we will explain some of the questions and relationships that arise in looking at integer sequences from these two perspectives. One of these connections occurs between prime numbers and closed orbits. In algebraic and geometric settings, there are hints of a quite widespread Pólya–Carlson dichotomy.

The following relation between Fibonacci and Lucas numbers of order k,$\sum _{i=0}^n{m}^i[l_i^{\left(k\right)}+(m-2)F_{i+1}^{\left(k\right)}-\sum _{j=3}^k(j-2\left)F_{i-j+1}^{\left(k\right)}\right]=m^{n+1}{F}_{n+1}^{\left(k\right)}+k-2,$ is derived by means of colored tiling. This relation generalizes the well-known Fibonacci-Lucas identities, ${\sum }_{i=0}^n{2}^i{L}_i=2^{n+1}{F}_{n+1},{\sum }_{i=0}^n{3}^i(L_i+F_{i+1})=3^{n+1}{F}_{n+1}$ and ${\sum }_{i=0}^n{m}^i(L_i+(m-2\left)F_{i+1}\right)=m^{n+1}{F}_{n+1}$ of A.T. Benjamin and J.J. Quinn, D. Marques, and T. Edgar, respectively.