scholarly journals The Least Eigenvalue of the Complement of the Square Power Graph of G

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Lubna Gul ◽  
Gohar Ali ◽  
Usama Waheed ◽  
Sumiya Nasir
Keyword(s):  
Positive Integer ◽  
Power Graph ◽  
Least Eigenvalue ◽  
The Family

Let G n , m represent the family of square power graphs of order n and size m , obtained from the family of graphs F n , k of order n and size k , with m ≥ k . In this paper, we discussed the least eigenvalue of graph G in the family G n , m c . All graphs considered here are undirected, simple, connected, and not a complete K n for positive integer n .

scholarly journals Least eigenvalue of the connected graphs whose complements are cacti

2019 ◽  
Vol 17 (1) ◽  
pp. 1319-1331
Author(s):  
Haiying Wang ◽  
Muhammad Javaid ◽  
Sana Akram ◽  
Muhammad Jamal ◽  
Shaohui Wang
Keyword(s):  
Linear Algebra ◽  
Adjacency Matrix ◽  
Connected Graphs ◽  
Least Eigenvalue ◽  
The Family

Abstract Suppose that Γ is a graph of order n and A(Γ) = [ai,j] is its adjacency matrix such that ai,j is equal to 1 if vi is adjacent to vj and ai,j is zero otherwise, where 1 ≤ i, j ≤ n. In a family of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix is minimum in the set of the least eigenvalues of all the graphs. Petrović et al. [On the least eigenvalue of cacti, Linear Algebra Appl., 2011, 435, 2357-2364] characterized a minimizing graph in the family of all cacti such that the complement of this minimizing graph is disconnected. In this paper, we characterize the minimizing graphs G ∈ $\begin{array}{} {\it\Omega}^c_n \end{array}$, i.e. $$\begin{array}{} \displaystyle \lambda_{min}(G)\leq\lambda_{min}(C^c) \end{array}$$ for each Cc ∈ $\begin{array}{} {\it\Omega}^c_n \end{array}$, where $\begin{array}{} {\it\Omega}^c_n \end{array}$ is a collection of connected graphs such that the complement of each graph of order n is a cactus with the condition that either its each block is only an edge or it has at least one block which is an edge and at least one block which is a cycle.

From the Outside In

2015 ◽  
Author(s):  
Derek Smith
Keyword(s):  
Positive Integer ◽  
Infinite Family ◽  
Open Problems ◽  
The Family

This chapter discusses Slothouber–Graatsma–Conway puzzle, which asks one to assemble six 1 × 2 × 2 pieces and three 1 × 1 × 1 pieces into the shape of a 3 × 3 × 3 cube. The puzzle has been generalized to larger cubes, and there is an infinite family of such puzzles. The chapter's primary argument is that, for any odd positive integer n = 2k + 1, there is exactly one way, up to symmetry, to make an n × n × n cube out of n tiny 1 × 1 × 1 cubes and six of each of a set of rectangular blocks. The chapter describes a way to solve each puzzle in the family and explains why there are no other solutions. It then presents several related open problems.

scholarly journals ON THE FAMILY OF DIOPHANTINE TRIPLES {k− 1,k+ 1, 16k3− 4k}

2007 ◽  
Vol 49 (2) ◽  
pp. 333-344 ◽  
Author(s):  
YANN BUGEAUD ◽  
ANDREJ DUJELLA ◽  
MAURICE MIGNOTTE
Keyword(s):  
Positive Integer ◽  
Recent Result ◽  
The Family ◽  
Diophantine Triples ◽  
Image Position ◽  
Diophantine Quadruples

AbstractIt is proven that ifk≥ 2 is an integer anddis a positive integer such that the product of any two distinct elements of the setincreased by 1 is a perfect square, thend= 4kord= 64k5−48k3+8k. Together with a recent result of Fujita, this shows that all Diophantine quadruples of the form {k− 1,k+ 1,c,d} are regular.

scholarly journals A Note on an Expressiveness Hierarchy for Multi-exit Iteration

2002 ◽  
Vol 9 (40) ◽  
Author(s):  
Luca Aceto ◽  
Willem Jan Fokkink ◽  
Anna Ingólfsdóttir
Keyword(s):  
Positive Integer ◽  
Process Algebra ◽  
Expressive Language ◽  
Basic Process ◽  
Star Operation ◽  
The Family ◽  
Bisimulation Equivalence

Multi-exit iteration is a generalization of the standard binary Kleene star operation that allows for the specification of agents that, up to bisimulation equivalence, are solutions of systems of recursion equations of the form<br />X_1 = P_1 X_2 + Q_1 <br /><br /> X_n = P_n X_1 + Q_n <br /><br /> where n is a positive integer, and the P_i and the Q_i are process terms. The addition of multi-exit iteration to Basic Process Algebra (BPA) yields a more expressive language than that obtained by augmenting BPA with the standard binary Kleene star. This note offers an expressiveness hierarchy, modulo bisimulation equivalence, for the family of multi-exit iteration operators proposed by Bergstra, Bethke and Ponse.

scholarly journals Vertex Isoperimetric Inequalities for a Family of Graphs on $\mathbb{Z}^k$

10.37236/2426 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Ellen Veomett ◽  
Andrew John Radcliffe
Keyword(s):  
Positive Integer ◽  
Isoperimetric Inequality ◽  
Isoperimetric Inequalities ◽  
The Family ◽  
Vertex Set

We consider the family of graphs whose vertex set is $\mathbb{Z}^k$ where two vertices are connected by an edge when their $\ell_\infty$-distance is 1.  We prove the optimal vertex isoperimetric inequality for this family of graphs.  That is, given a positive integer $n$, we find a set $A\subset \mathbb{Z}^k$ of size $n$ such that the number of vertices who share an edge with some vertex in $A$ is minimized.  These sets of minimal boundary  are nested, and the proof uses the technique of compression.We also show a method of calculating the vertex boundary for certain subsets in  this family of graphs.  This calculation and the isoperimetric inequality allow us to indirectly find the sets which minimize the function calculating the boundary.

scholarly journals Uniform congruence counting for Schottky semigroups in SL2(𝐙)

2019 ◽  
Vol 2019 (753) ◽  
pp. 89-135 ◽  
Author(s):  
Michael Magee ◽  
Hee Oh ◽  
Dale Winter
Keyword(s):  
Hausdorff Dimension ◽  
Positive Integer ◽  
Continued Fractions ◽  
Euclidean Norm ◽  
Limit Set ◽  
The Family ◽  
Prime Factors ◽  
Uniform Congruence

AbstractLet Γ be a Schottky semigroup in {\mathrm{SL}_{2}(\mathbf{Z})}, and for {q\in\mathbf{N}}, let{\Gamma(q):=\{\gamma\in\Gamma:\gamma=e~{}(\mathrm{mod}~{}q)\}}be its congruence subsemigroup of level q. Let δ denote the Hausdorff dimension of the limit set of Γ. We prove the following uniform congruence counting theorem with respect to the family of Euclidean norm balls {B_{R}} in {M_{2}(\mathbf{R})} of radius R: for all positive integer q with no small prime factors,\#(\Gamma(q)\cap B_{R})=c_{\Gamma}\frac{R^{2\delta}}{\#(\mathrm{SL}_{2}(% \mathbf{Z}/q\mathbf{Z}))}+O(q^{C}R^{2\delta-\epsilon})as {R\to\infty} for some {c_{\Gamma}>0,C>0,\epsilon>0} which are independent of q. Our technique also applies to give a similar counting result for the continued fractions semigroup of {\mathrm{SL}_{2}(\mathbf{Z})}, which arises in the study of Zaremba’s conjecture on continued fractions.

Ulam stability of some functional inclusions for multi-valued mappings

Filomat ◽  
2017 ◽  
Vol 31 (17) ◽  
pp. 5489-5495 ◽  
Author(s):  
Janusz Brzdęk ◽  
Magdalena Piszczek
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Positive Integer ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Main Tool ◽  
Ulam Stability ◽  
Nonlinear Operators ◽  
The Family ◽  
Fixed Positive Integer

We show that some multifunctions F : K ? n(Y), satisfying functional inclusions of the form ? (x,F(?1(x)),..., F(?n(x)))? F(x)G(x), admit near-selections f : K ? Y, fulfilling the functional equation ? (x,f (?1(x)),..,, f(?n(x)))= f(x), where functions G : K ? n(Y), ?: K x Yn ? Y and ?1,..., ?n ? KK are given, n is a fixed positive integer, K is a nonempty set, (Y,?) is a group and n(Y) denotes the family of all nonempty subsets of Y. Our results have been motivated by the notion of Ulam stability and some earlier outcomes. The main tool in the proofs is a very recent fixed point theorem for nonlinear operators, acting on some spaces of multifunctions.

scholarly journals Generalized Mutual Information

Stats ◽  
2020 ◽  
Vol 3 (2) ◽  
pp. 158-165
Author(s):  
Zhiyi Zhang
Keyword(s):  
Information Theory ◽  
Positive Integer ◽  
Mutual Information ◽  
General Class ◽  
Building Blocks ◽  
Potential Utility ◽  
First Order ◽  
The Family ◽  
Random Elements

Mutual information is one of the essential building blocks of information theory. It is however only finitely defined for distributions in a subclass of the general class of all distributions on a joint alphabet. The unboundedness of mutual information prevents its potential utility from being extended to the general class. This is in fact a void in the foundation of information theory that needs to be filled. This article proposes a family of generalized mutual information whose members are indexed by a positive integer n, with the nth member being the mutual information of nth order. The mutual information of the first order coincides with Shannon’s, which may or may not be finite. It is however established (a) that each mutual information of an order greater than 1 is finitely defined for all distributions of two random elements on a joint countable alphabet, and (b) that each and every member of the family enjoys all the utilities of a finite Shannon’s mutual information.

scholarly journals An Equational Axiomatization for Multi-Exit Iteration

1996 ◽  
Vol 3 (22) ◽  
Author(s):  
Luca Aceto ◽  
Willem Jan Fokkink
Keyword(s):  
Positive Integer ◽  
Process Algebra ◽  
Expressive Language ◽  
General Structure ◽  
Basic Process ◽  
Star Operation ◽  
The Family ◽  
Bisimulation Equivalence

<p>This paper presents an equational axiomatization of bisimulation equivalence over the language of Basic Process Algebra (BPA) with multi-exit iteration. Multi-exit iteration is a generalization of the standard binary Kleene star operation that allows for the specification of agents that, up to bisimulation equivalence, are solutions<br />of systems of recursion equations of the form</p><p>X1 = P1 X2 + Q1<br />...<br />Xn = Pn X1 + Qn</p><p>where n is a positive integer, and the Pi and the Qi are process terms. The addition<br />of multi-exit iteration to BPA yields a more expressive language than that obtained by augmenting BPA with the standard binary Kleene star (BPA). As a<br />consequence, the proof of completeness of the proposed equational axiomatization<br />for this language, although standard in its general structure, is much more involved<br />than that for BPA. An expressiveness hierarchy for the family of k-exit iteration operators proposed by Bergstra, Bethke and Ponse is also offered.</p><p> </p>

CUTTING AND PASTING OF RIEMANN SURFACES WITH ABELIAN DIFFERENTIALS I

1999 ◽  
Vol 10 (05) ◽  
pp. 587-617
Author(s):  
YOSHITAKE HASHIMOTO ◽  
KIYOSHI OHBA
Keyword(s):  
Riemann Surface ◽  
Positive Integer ◽  
Elliptic Curves ◽  
Complex Plane ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Complex Structures ◽  
Line Segments ◽  
Abelian Differentials ◽  
The Family

We introduce a method of constructing once punctured Riemann surfaces by cutting the complex plane along "line segments" and pasting by "parallel transformations". The advantage of this construction is to give a good visualization of the deformation of complex structures of Riemann surfaces. In fact, given a positive integer g, there appears a family of once punctured Riemann surfaces of genus g which is complete and effectively parametrized at any point. Our construction naturally gives each of the resulting surfaces what we call a Lagrangian lattice Λ, a certain subgroup of the first homology. Furthermore Λ and the puncture determine an Abelian differential ωΛ of the second kind on the Riemann surface. Using Λ and ωΛ we consider the Kodaira–Spencer maps and some extension of the family to obtain any once punctured Riemann surface with a Lagrangian lattice. In particular we describe the moduli space of once punctured elliptic curves with Lagrangian lattices.

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