scholarly journals On distances in generalized Sierpiński graphs

2018 ◽  
Vol 12 (1) ◽  
pp. 49-69 ◽  
Author(s):  
Alejandro Estrada-Moreno ◽  
Erick Rodríguez-Bazan ◽  
Juan Rodríguez-Velázquez
Keyword(s):  
Explicit Formula ◽  
Recursive Formula ◽  
Arbitrary Vertex ◽  
Base Graph ◽  
Sierpiński Graphs ◽  
Recursive Formulas

In this paper we propose formulas for the distance between vertices of a generalized Sierpi?ski graph S(G, t) in terms of the distance between vertices of the base graph G. In particular, we deduce a recursive formula for the distance between an arbitrary vertex and an extreme vertex of S(G, t), and we obtain a recursive formula for the distance between two arbitrary vertices of S(G, t) when the base graph is triangle-free. From these recursive formulas, we provide algorithms to compute the distance between vertices of S(G, t). In addition, we give an explicit formula for the diameter and radius of S(G, t) when the base graph is a tree.

scholarly journals On distances in Sierpiński graphs: Almost-extreme vertices and metric dimension

2013 ◽  
Vol 7 (1) ◽  
pp. 72-82 ◽  
Author(s):  
Sandi Klavzar ◽  
Sara Zemljic
Keyword(s):  
Computer Science ◽  
Metric Dimension ◽  
Fractal Nature ◽  
Explicit Formulas ◽  
Arbitrary Vertex ◽  
Tower Of Hanoi ◽  
Sierpiński Graphs ◽  
Total Distance

Sierpi?ski graphs Sn p form an extensively studied family of graphs of fractal nature applicable in topology, mathematics of the Tower of Hanoi, computer science, and elsewhere. An almost-extreme vertex of Sn p is introduced as a vertex that is either adjacent to an extreme vertex of Sn p or is incident to an edge between two subgraphs of Sn p isomorphic to Snp-1. Explicit formulas are given for the distance in Sn p between an arbitrary vertex and an almostextreme vertex. The formulas are applied to compute the total distance of almost-extreme vertices and to obtain the metric dimension of Sierpi?ski graphs.

scholarly journals Counting Derangements, Non Bijective Functions and the Birthday Problem

2010 ◽  
Vol 18 (4) ◽  
pp. 197-200
Author(s):  
Cezary Kaliszyk
Keyword(s):  
Explicit Formula ◽  
Recursive Formula ◽  
Finite Sets ◽  
Birthday Problem ◽  
Finite Set ◽  
One To One

Counting Derangements, Non Bijective Functions and the Birthday Problem The article provides counting derangements of finite sets and counting non bijective functions. We provide a recursive formula for the number of derangements of a finite set, together with an explicit formula involving the number e. We count the number of non-one-to-one functions between to finite sets and perform a computation to give explicitely a formalization of the birthday problem. The article is an extension of [10].

scholarly journals On the Stability of Recursive Formulas

1993 ◽  
Vol 23 (2) ◽  
pp. 227-258 ◽  
Author(s):  
Harry H. Panjer ◽  
Shaun Wang
Keyword(s):  
Binomial Distribution ◽  
Negative Binomial ◽  
Recursive Formula ◽  
Absolute Error ◽  
Recurrence Equation ◽  
Simple Method ◽  
Numerical Examples ◽  
Compound Binomial ◽  
Recursive Formulas ◽  
The Stability

AbstractBased on recurrence equation theory and relative error (rather than absolute error) analysis, the concept and criterion for the stability of a recurrence equation are clarified. A family of recursions, called congruent recursions, is proved to be strongly stable in evaluating its non-negative solutions. A type of strongly unstable recursion is identified. The recursive formula discussed by Panjer (1981) is proved to be strongly stable in evaluating the compound Poisson and the compound Negative Binomial (including Geometric) distributions. For the compound Binomial distribution, the recursion is shown to be unstable. A simple method to cope with this instability is proposed. Many other recursions are reviewed. Illustrative numerical examples are given.

scholarly journals Several Explicit and Recursive Formulas for the Generalized Motzkin Numbers

2017 ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Recursive Formula ◽  
Catalan Numbers ◽  
Explicit Formulas ◽  
Motzkin Numbers ◽  
Recursive Formulas

In the paper, the authors find two explicit formulas and recover a recursive formula for the generalized Motzkin numbers. Consequently, the authors deduce two explicit formulas and a recursive formula for the Motzkin numbers, the Catalan numbers, and the restricted hexagonal numbers respectively.

A multi-parameter generalization of the symmetric algorithm

2018 ◽  
Vol 68 (4) ◽  
pp. 699-712
Author(s):  
José L. Ramírez ◽  
Mark Shattuck
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Fibonacci Numbers ◽  
Recursive Formula ◽  
Binomial Coefficient ◽  
Fibonacci Polynomials ◽  
Combinatorial Argument ◽  
Recurrent Sequences ◽  
Seidel Method ◽  
Multivariate Generalization

Abstract The symmetric algorithm is a variant of the well-known Euler-Seidel method which has proven useful in the study of linearly recurrent sequences. In this paper, we introduce a multivariate generalization of the symmetric algorithm which reduces to it when all parameters are unity. We derive a general explicit formula via a combinatorial argument and also an expression for the row generating function. Several applications of our algorithm to the q-Fibonacci and q-hyper-Fibonacci numbers are discussed. Among our results is an apparently new recursive formula for the Carlitz Fibonacci polynomials. Finally, a (p, q)-analogue of the algorithm is introduced and an explicit formula for it in terms of the (p, q)-binomial coefficient is found.

scholarly journals Recognizing generalized Sierpiński graphs

2020 ◽  
Vol 14 (1) ◽  
pp. 122-137
Author(s):  
Wilfried Imrich ◽  
Iztok Peterin
Keyword(s):  
Polynomial Time ◽  
Complete Graph ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Arbitrary Graph ◽  
Base Graph ◽  
Sierpiński Graphs ◽  
Vertex Set

Let H be an arbitrary graph with vertex set V (H) = [nH] = {l,?, nH}. The generalized Sierpi?ski graph SnH , n ? N, is defined on the vertex set [nH]n, two different vertices u = un ?u1 and v = vn ? v1 being adjacent if there exists an h? [n] such that (a) ut = vt, for t > h, (b) uh ? vh and uhvh ? E(H), and (c) ut = vh and vt = uh for t < h. If H is the complete graph Kk, then we speak of the Sierpi?ski graph Sn k . We present an algorithm that recognizes Sierpi?ski graphs Sn k in O(|V (Sn k )|1+1=n) = O(|E(Sn k )|) time. For generalized Sierpi?ski graphs SnH we present a polynomial time algorithm for the case when H belong to a certain well defined class of graphs. We also describe how to derive the base graph H from an arbitrarily given SnH .

scholarly journals On the roman domination number of generalized Sierpiński graphs

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6515-6528 ◽  
Author(s):  
F. Ramezani ◽  
E.D. Rodríguez-Bazan ◽  
J.A. Rodríguez-Velázquez
Keyword(s):  
Complete Graph ◽  
Upper Bound ◽  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Base Graph ◽  
Sierpiński Graphs ◽  
Roman Domination Number ◽  
Roman Dominating Function ◽  
Dominating Function

A map f : V?(0,1,2) is a Roman dominating function on a graph G = (V,E) if for every vertex v ? V with f(v)=0, there exists a vertex u, adjacent to v, such that f(u)=2. The weight of a Roman dominating function is given by f(V)=?u?V f(u). The minimum weight among all Roman dominating functions on G is called the Roman domination number of G. In this article we study the Roman domination number of Generalized Sierpi?ski graphs S(G,t). More precisely, we obtain a general upper bound on the Roman domination number of S(G,t) and discuss the tightness of this bound. In particular, we focus on the cases in which the base graph G is a path, a cycle, a complete graph or a graph having exactly one universal vertex.

scholarly journals Differential equations and expansions for quaternionic modular forms in the discriminant 6 case

2012 ◽  
Vol 15 ◽  
pp. 385-399 ◽  
Author(s):  
Srinath Baba ◽  
Håkan Granath
Keyword(s):  
Differential Equations ◽  
Explicit Formula ◽  
Modular Forms ◽  
Quaternion Algebra ◽  
Recursive Formula ◽  
Unit Group ◽  
Rational Structure ◽  
Taylor Coefficients ◽  
Taylor Expansions ◽  
Modular Polynomials

AbstractWe study the differential structure of the ring of modular forms for the unit group of the quaternion algebra over ℚ of discriminant 6. Using these results we give an explicit formula for Taylor expansions of the modular forms at the elliptic points. Using appropriate normalizations we show that the Taylor coefficients at the elliptic points of the generators of the ring of modular forms are all rational and 6-integral. This gives a rational structure on the ring of modular forms. We give a recursive formula for computing the Taylor coefficients of modular forms at elliptic points and, as an application, give an algorithm for computing modular polynomials.

Precise Capacity Analysis in Binary Networks with Multiple Coding Level Inputs

2010 ◽  
Vol 22 (3) ◽  
pp. 660-688 ◽  
Author(s):  
Yali Amit ◽  
Yibi Huang
Keyword(s):  
Markov Chain ◽  
Explicit Formula ◽  
Hebbian Learning ◽  
Normal Approximation ◽  
Recursive Formula ◽  
Total Activity ◽  
Neural Dynamics ◽  
Capacity Analysis ◽  
Initial State ◽  
Binary Networks

We compute retrieval probabilities as a function of pattern age for networks with binary neurons and synapses updated with the simple Hebbian learning model studied in Amit and Fusi ( 1994 ). The analysis depends on choosing a neural threshold that enables patterns to stabilize in the neural dynamics. In contrast to most earlier work, where selective neurons for each pattern are drawn independently with fixed probability f, here we analyze the situation where f is drawn from some distribution on a range of coding levels. In order to set a workable threshold in this setting, it is necessary to introduce a simple inhibition in the neural dynamics whose magnitude depends on the total activity of the network. Proper choice of the threshold depends on the value of the covariances between the synapses for which we provide an explicit formula. Retrieval probabilities depend on the distribution of the fields induced by a learned pattern. We show that the field induced by the first learned pattern evolves as a Markov chain during subsequent learning epochs, leading to a recursive formula for the distribution. Alternatively, the distribution can be computed using a normal approximation, which involves the value of the synaptic covariances. Capacity is computed as the sum of the retrival probabilities over all ages. We show through simulation that the chosen threshold enables retrieval with asynchronous dynamics even in the presence of significant noise in the initial state of the pattern. The computed probabilities with both methods are shown to be very close to probabilities estimated from simulation. The analysis is extended to randomly connected networks.

scholarly journals Connectivity and some other properties of generalized Sierpiński graphs

2018 ◽  
Vol 12 (2) ◽  
pp. 401-412
Author(s):  
Sandi Klavzar ◽  
Sara Zemljic
Keyword(s):  
Positive Integer ◽  
Explicit Formula ◽  
Complete Graph ◽  
Building Block ◽  
Connected Components ◽  
Edge Connectivity ◽  
Sierpiński Graphs

If G is a graph and n a positive integer, then the generalized Sierpi?ski graph SnG is a fractal-like graph that uses G as a building block. The construction of SnG generalizes the classical Sierpi?ski graphs Sn p, where the role of G is played by the complete graph Kp. An explicit formula for the number of connected components in SnG is given and it is proved that the (edge-)connectivity of SnG equals the (edge-)connectivity of G. It is demonstrated that SnG contains a 1-factor if and only if G contains a 1-factor. Hamiltonicity of generalized Sierpi?ski graphs is also discussed.

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