Geometric Proof of the Tetrahedral Number Formula

2007 ◽  
Keyword(s):  
Geometric Proof ◽  
Number Formula

Bounds on hyper-Wiener index of graphs

2017 ◽  
Vol 10 (03) ◽  
pp. 1750057
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Sadegh Rahimi
Keyword(s):  
Lower Bounds ◽  
Organic Molecules ◽  
Wiener Index ◽  
Upper And Lower Bounds ◽  
Pharmacological Properties ◽  
Graph Theoretic ◽  
Acyclic Graphs ◽  
Wiener Number ◽  
Number Formula ◽  
Cyclic Graphs

The Wiener number [Formula: see text] of a graph [Formula: see text] was introduced by Harold Wiener in connection with the modeling of various physic-chemical, biological and pharmacological properties of organic molecules in chemistry. Milan Randić introduced a modification of the Wiener index for trees (acyclic graphs), and it is known as the hyper-Wiener index. Then Klein et al. generalized Randić’s definition for all connected (cyclic) graphs, as a generalization of the Wiener index, denoted by [Formula: see text] and defined as [Formula: see text]. In this paper, we establish some upper and lower bounds for [Formula: see text], in terms of other graph-theoretic parameters. Moreover, we compute hyper-Wiener number of some classes of graphs.

A Geometric Proof of lim d →0 +(-dln d) = 0

1992 ◽  
Vol 23 (3) ◽  
pp. 209
Author(s):  
John H. Mathews
Keyword(s):  
Geometric Proof

scholarly journals On S-class number relations of algebraic tori in Galois extensions of global fields

1991 ◽  
Vol 124 ◽  
pp. 133-144 ◽  
Author(s):  
Masanori Morishita
Keyword(s):  
Class Number ◽  
Quadratic Forms ◽  
Euler Number ◽  
Number Fields ◽  
Binary Quadratic Forms ◽  
Algebraic Number Fields ◽  
Algebraic Tori ◽  
Genus Theory ◽  
Class Number Formula ◽  
Number Formula

As an interpretation and a generalization of Gauss’ genus theory on binary quadratic forms in the language of arithmetic of algebraic tori, Ono [02] established an equality between a kind of “Euler number E(K/k)” for a finite Galois extension K/k of algebraic number fields and other arithmetical invariants associated to K/k. His proof depended on his Tamagawa number formula [01] and Shyr’s formula [Sh] which follows from the analytic class number formula of a torus. Later, two direct proofs were given by Katayama [K] and Sasaki [Sa].

A Lattice-Geometric Proof of Wedderburn’s Theorem

1993 ◽  
Vol 23 (3-4) ◽  
pp. 384-386 ◽  
Author(s):  
Stefan E. Schmidt
Keyword(s):  
Geometric Proof

Roman domination and 2-independence in trees

2017 ◽  
Vol 09 (02) ◽  
pp. 1750023 ◽  
Author(s):  
Nacéra Meddah ◽  
Mustapha Chellali
Keyword(s):  
Independence Number ◽  
Minimum Weight ◽  
Domination Number ◽  
Independent Set ◽  
Maximum Cardinality ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Number Formula ◽  
Sum Formula ◽  
Roman Dominating Function

A Roman dominating function (RDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying the condition that every vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] of [Formula: see text] for which [Formula: see text]. The weight of a RDF is the sum [Formula: see text], and the minimum weight of a RDF [Formula: see text] is the Roman domination number [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is a [Formula: see text]-independent set of [Formula: see text] if every vertex of [Formula: see text] has at most one neighbor in [Formula: see text] The maximum cardinality of a [Formula: see text]-independent set of [Formula: see text] is the [Formula: see text]-independence number [Formula: see text] Both parameters are incomparable in general, however, we show that if [Formula: see text] is a tree, then [Formula: see text]. Moreover, all extremal trees attaining equality are characterized.

scholarly journals A characterization of the weighted version of McEliece–Rodemich–Rumsey–Schrijver number based on convex quadratic programming

2015 ◽  
Vol 07 (04) ◽  
pp. 1550050
Author(s):  
Carlos J. Luz
Keyword(s):  
Quadratic Programming ◽  
Discrete Math ◽  
Convex Quadratic Programming ◽  
Stability Number ◽  
Weighted Version ◽  
Similar Characterization ◽  
Number Formula ◽  
Weighted Stability ◽  
Theta Number

For any graph [Formula: see text] Luz and Schrijver [A convex quadratic characterization of the Lovász theta number, SIAM J. Discrete Math. 19(2) (2005) 382–387] introduced a characterization of the Lovász number [Formula: see text] based on convex quadratic programming. A similar characterization is now established for the weighted version of the number [Formula: see text] independently introduced by McEliece, Rodemich, and Rumsey [The Lovász bound and some generalizations, J. Combin. Inform. Syst. Sci. 3 (1978) 134–152] and Schrijver [A Comparison of the Delsarte and Lovász bounds, IEEE Trans. Inform. Theory 25(4) (1979) 425–429]. Also, a class of graphs for which the weighted version of [Formula: see text] coincides with the weighted stability number is characterized.

IWASAWA TYPE FORMULA FOR COVERS OF A LINK IN A RATIONAL HOMOLOGY SPHERE

2008 ◽  
Vol 17 (10) ◽  
pp. 1199-1221 ◽  
Author(s):  
TERUHISA KADOKAMI ◽  
YASUSHI MIZUSAWA
Keyword(s):  
Number Field ◽  
Class Number ◽  
Homology Sphere ◽  
Rational Homology ◽  
Homology Groups ◽  
Cyclic Covers ◽  
Class Number Formula ◽  
Number Formula ◽  
Iwasawa Invariants ◽  
Rational Homology Sphere

Based on the analogy between links and primes, we present an analogue of the Iwasawa's class number formula in a Zp-extension for the p-homology groups of pn-fold cyclic covers of a link in a rational homology 3-sphere. We also describe the associated Iwasawa invariants precisely for some examples and discuss analogies with the number field case.

Finiteness of the set of virtual knots with a given state number

2018 ◽  
Vol 27 (08) ◽  
pp. 1850049
Author(s):  
Takuji Nakamura ◽  
Yasutaka Nakanishi ◽  
Shin Satoh
Keyword(s):  
The Real ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Minimum Number ◽  
Number Formula

A state of a virtual knot diagram [Formula: see text] is a collection of circles obtained from [Formula: see text] by splicing all the real crossings. For each integer [Formula: see text], we denote by [Formula: see text] the number of states of [Formula: see text] with [Formula: see text] circles. The [Formula: see text]-state number [Formula: see text] of a virtual knot [Formula: see text] is the minimum number of [Formula: see text] for [Formula: see text] of [Formula: see text]. Let [Formula: see text] be the set of virtual knots [Formula: see text] with [Formula: see text] for an integer [Formula: see text]. In this paper, we study the finiteness of [Formula: see text]. We determine the finiteness of [Formula: see text] for any [Formula: see text] and [Formula: see text] for any [Formula: see text].

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