scholarly journals Biembeddings of Metacyclic Groups and Triangulations of Orientable Surfaces by Complete Graphs

10.37236/2169 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Michael John Grannell ◽  
Martin Knor
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Complete Graph ◽  
Latin Squares ◽  
Orientable Surface ◽  
Complete Graphs ◽  
Metacyclic Group ◽  
Metacyclic Groups ◽  
Multiplicative Order ◽  
Orientable Surfaces

For each integer $n\ge 3$, $n\ne 4$, for each odd integer $m\ge 3$, and for any $\lambda\in\Bbb Z_n$ of (multiplicative) order $m'$ where $m'\mid m$, we construct a biembedding of Latin squares in which one of the squares is the Cayley table of the metacyclic group $\mathbb{Z}_m\ltimes_{\lambda}\mathbb{Z}_n$. This extends the spectrum of Latin squares known to be biembeddable.The best existing lower bounds for the number of triangular embeddings of a complete graph $K_z$ in an orientable surface are of the form $z^{z^2(a-o(1))}$ for suitable positive constants $a$ and for restricted infinite classes of $z$. Using embeddings of $\mathbb{Z}_3\ltimes_{\lambda}\mathbb{Z}_n$, we extend this lower bound to a substantially larger class of values of $z$.

New bounds of Mutually unbiased maximally entangled bases in C^dxC^(kd)

2018 ◽  
Vol 18 (13&14) ◽  
pp. 1152-1164
Author(s):  
Xiaoya Cheng ◽  
Yun Shang
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Prime Power ◽  
Latin Squares ◽  
Mutually Unbiased Bases ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Bipartite System

Mutually unbiased bases which is also maximally entangled bases is called mutually unbiased maximally entangled bases (MUMEBs). We study the construction of MUMEBs in bipartite system. In detail, we construct 2(p^a-1) MUMEBs in \cd by properties of Guss sums for arbitrary odd d. It improves the known lower bound p^a-1 for odd d. Certainly, it also generalizes the lower bound 2(p^a-1) for d being a single prime power. Furthermore, we construct MUMEBs in \ckd for general k\geq 2 and odd d. We get the similar lower bounds as k,b are both single prime powers. Particularly, when k is a square number, by using mutually orthogonal Latin squares, we can construct more MUMEBs in \ckd, and obtain greater lower bounds than reducing the problem into prime power dimension in some cases.

scholarly journals LOWER BOUNDS FOR PERIODS OF DUCCI SEQUENCES

2019 ◽  
Vol 102 (1) ◽  
pp. 31-38
Author(s):  
FLORIAN BREUER ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Multiplicative Order ◽  
Image Position

A Ducci sequence is a sequence of integer $n$-tuples obtained by iterating the map $$\begin{eqnarray}D:(a_{1},a_{2},\ldots ,a_{n})\mapsto (|a_{1}-a_{2}|,|a_{2}-a_{3}|,\ldots ,|a_{n}-a_{1}|).\end{eqnarray}$$ Such a sequence is eventually periodic and we denote by $P(n)$ the maximal period of such sequences for given odd $n$. We prove a lower bound for $P(n)$ by counting certain partitions. We then estimate the size of these partitions via the multiplicative order of two modulo $n$.

IMPROVED LOWER BOUND FOR THE NUMBER OF KNOTTED HAMILTONIAN CYCLES IN SPATIAL EMBEDDINGS OF COMPLETE GRAPHS

2010 ◽  
Vol 19 (05) ◽  
pp. 705-708 ◽  
Author(s):  
YOSHIYASU HIRANO
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Hamiltonian Cycles ◽  
Complete Graphs

We prove that every spatial embedding of the complete graph K8 contains at least 3 knotted Hamiltonian cycles, and that every spatial embedding of Kn contains at least 3(n - 1)(n - 2) ⋯ 8 knotted Hamiltonian cycles, for n > 8.

scholarly journals $k$-Cycle Free One-Factorizations of Complete Graphs

10.37236/92 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Mariusz Meszka
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Complete Graphs

It is proved that for every $n\geq 3$ and every even $k\geq 4$, where $k\neq 2n$, there exists one-factorization of the complete graph $K_{2n}$ such that any two one-factors do not induce a graph with a cycle of length $k$ as a component. Moreover, some infinite classes of one-factorizations, in which lengths of cycles induced by any two one-factors satisfy a given lower bound, are constructed.

scholarly journals Congruence Classes of Orientable $2$-Cell Embeddings of Bouquets of Circles and Dipoles

10.37236/313 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Yan-Quan Feng ◽  
Jin-Ho Kwak ◽  
Jin-Xin Zhou
Keyword(s):  
Orientable Surface ◽  
Simple Graph ◽  
Complete Graphs ◽  
Graph Automorphism ◽  
Wheel Graphs ◽  
Complete Bipartite Graphs ◽  
Multiple Edges ◽  
Orientable Surfaces ◽  
Surface Homeomorphism ◽  
Congruence Classes

Two $2$-cell embeddings $\imath : X \to S$ and $\jmath : X \to S$ of a connected graph $X$ into a closed orientable surface $S$ are congruent if there are an orientation-preserving surface homeomorphism $h : S \to S$ and a graph automorphism $\gamma$ of $X$ such that $\imath h =\gamma\jmath$. Mull et al. [Proc. Amer. Math. Soc. 103(1988) 321–330] developed an approach for enumerating the congruence classes of $2$-cell embeddings of a simple graph (without loops and multiple edges) into closed orientable surfaces and as an application, two formulae of such enumeration were given for complete graphs and wheel graphs. The approach was further developed by Mull [J. Graph Theory 30(1999) 77–90] to obtain a formula for enumerating the congruence classes of $2$-cell embeddings of complete bipartite graphs into closed orientable surfaces. By considering automorphisms of a graph as permutations on its dart set, in this paper Mull et al.'s approach is generalized to any graph with loops or multiple edges, and by using this method we enumerate the congruence classes of $2$-cell embeddings of a bouquet of circles and a dipole into closed orientable surfaces.

A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers

10.37236/1188 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Geoffrey Exoo
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Ramsey Numbers

For $k \geq 5$, we establish new lower bounds on the Schur numbers $S(k)$ and on the k-color Ramsey numbers of $K_3$.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

scholarly journals On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 512
Author(s):  
Maryam Baghipur ◽  
Modjtaba Ghorbani ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Distance Matrix ◽  
Upper And Lower Bounds ◽  
Laplacian Eigenvalue ◽  
Signless Laplacian ◽  
Graph Parameters ◽  
Simple Connected Graph ◽  
Second Largest Eigenvalue ◽  
Reciprocal Distance Matrix

The signless Laplacian reciprocal distance matrix for a simple connected graph G is defined as RQ(G)=diag(RH(G))+RD(G). Here, RD(G) is the Harary matrix (also called reciprocal distance matrix) while diag(RH(G)) represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with n vertices, the complete graph Kn and the graph Kn−e obtained from Kn by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue.

scholarly journals A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 164
Author(s):  
Tobias Rupp ◽  
Stefan Funke
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real World ◽  
Road Network ◽  
Upper And Lower Bounds ◽  
Bounded Degree ◽  
Shortest Path Routing ◽  
Graph Classes ◽  
Speed Up ◽  
To Come

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

scholarly journals On Computing Multilinear Polynomials Using Multi- r -ic Depth Four Circuits

2021 ◽  
Vol 13 (3) ◽  
pp. 1-21
Author(s):  
Suryajith Chillara
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Complexity Measure ◽  
Natural Question ◽  
Multilinear Polynomial ◽  
Arithmetic Circuit ◽  
Theory Of Computing ◽  
Small Constant ◽  
Multilinear Polynomials ◽  
The Individual

In this article, we are interested in understanding the complexity of computing multilinear polynomials using depth four circuits in which the polynomial computed at every node has a bound on the individual degree of r ≥ 1 with respect to all its variables (referred to as multi- r -ic circuits). The goal of this study is to make progress towards proving superpolynomial lower bounds for general depth four circuits computing multilinear polynomials, by proving better bounds as the value of r increases. Recently, Kayal, Saha and Tavenas (Theory of Computing, 2018) showed that any depth four arithmetic circuit of bounded individual degree r computing an explicit multilinear polynomial on n O (1) variables and degree d must have size at least ( n / r 1.1 ) Ω(√ d / r ) . This bound, however, deteriorates as the value of r increases. It is a natural question to ask if we can prove a bound that does not deteriorate as the value of r increases, or a bound that holds for a larger regime of r . In this article, we prove a lower bound that does not deteriorate with increasing values of r , albeit for a specific instance of d = d ( n ) but for a wider range of r . Formally, for all large enough integers n and a small constant η, we show that there exists an explicit polynomial on n O (1) variables and degree Θ (log 2 n ) such that any depth four circuit of bounded individual degree r ≤ n η must have size at least exp(Ω(log 2 n )). This improvement is obtained by suitably adapting the complexity measure of Kayal et al. (Theory of Computing, 2018). This adaptation of the measure is inspired by the complexity measure used by Kayal et al. (SIAM J. Computing, 2017).

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