Mathematical Combinatorics (International Book Series), vol. IV, 2019

2020·1.9 MB·English

by  Mao, Linfan

ISBN 978-1-59973-642-6
VOLUME 4, 2019
MATHEMATICAL COMBINATORICS
(INTERNATIONAL BOOK SERIES)
Edited By Linfan MAO
THE MADIS OF CHINESE ACADEMY OF SCIENCES AND
ACADEMY OF MATHEMATICAL COMBINATORICS & APPLICATIONS, USA
December, 2019
Vol.4, 2019 ISBN 978-1-59973-642-6
MATHEMATICAL COMBINATORICS
(INTERNATIONAL BOOK SERIES)
Edited By Linfan MAO
(www.mathcombin.com)
The Madis of Chinese Academy of Sciences and
Academy of Mathematical Combinatorics & Applications, USA
December, 2019
Aims and Scope: The mathematical combinatorics is a subject that applying combinatorial
notiontoallmathematicsandallsciencesforunderstandingtherealityofthingsintheuniverse,
motivatedbyCCConjectureofDr.LinfanMAOonmathematicalsciences. TheMathematical
Combinatorics (International Book Series) is a fully refereed international book series
withanISBNnumberoneachissue,sponsoredbytheMADIS of Chinese Academy of Sciences
andpublishedinUSAquarterly,whichpublishesoriginalresearchpapersandsurveyarticlesin
allaspectsofmathematicalcombinatorics,Smarandachemulti-spaces,Smarandachegeometries,
non-Euclidean geometry, topology and their applications to other sciences. Topics in detail to
be covered are:
Mathematical combinatorics;
Smarandachemulti-spacesandSmarandachegeometrieswithapplicationstoothersciences;
Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and
map enumeration; Combinatorial designs; Combinatorial enumeration;
Differential Geometry; Geometry on manifolds; Low Dimensional Topology; Differential
Topology; Topology of Manifolds;
Geometrical aspects of Mathematical Physics and Relations with Manifold Topology;
Mathematical theory on gravitationalfields and parallel universes;
Applications of Combinatorics to mathematics and theoretical physics.
Generally,papersonapplicationsofcombinatoricstoothermathematicsandothersciences
are welcome by this journal.
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Subscription A subscription can be ordered by an email directly to
Linfan Mao
The Editor-in-Chief of International Journal of Mathematical Combinatorics
Chinese Academy of Mathematics and System Science Beijing, 100190, P.R.China, and also the
President of Academy of Mathematical Combinatorics & Applications (AMCA), Colorado, USA
Email: [email protected]
Price: US$48.00
Editorial Board (4th)
Editor-in-Chief
Linfan MAO Shaofei Du
ChineseAcademyofMathematicsandSystem Capital Normal University, P.R.China
Science, P.R.China Email: [email protected]
and
Xiaodong Hu
Academy of Mathematical Combinatorics &
ChineseAcademyofMathematicsandSystem
Applications, Colorado,USA
Science, P.R.China
Email: [email protected]
Email: [email protected]
Deputy Editor-in-Chief
Yuanqiu Huang
Hunan Normal University, P.R.China
Guohua Song Email: [email protected]
Beijing University of Civil Engineering and
H.Iseri
Architecture, P.R.China
Mansfield University, USA
Email: [email protected]
Email: hiseri@mnsfld.edu
Editors Xueliang Li
Nankai University, P.R.China
Arindam Bhattacharyya Email: [email protected]
Jadavpur University, India Guodong Liu
Email: [email protected] Huizhou University
Said Broumi Email: [email protected]
Hassan II University Mohammedia W.B.Vasantha Kandasamy
Hay El Baraka Ben M’sik Casablanca Indian Institute of Technology, India
B.P.7951 Morocco Email: [email protected]
Junliang Cai Ion Patrascu
Beijing Normal University, P.R.China Fratii Buzesti National College
Email: [email protected] Craiova Romania
Yanxun Chang Han Ren
Beijing Jiaotong University, P.R.China East China Normal University, P.R.China
Email: [email protected] Email: [email protected]
Jingan Cui Ovidiu-Ilie Sandru
Beijing University of Civil Engineering and Politechnica University of Bucharest
Architecture, P.R.China Romania
Email: [email protected]
ii MathematicalCombinatorics(InternationalBookSeries)
Mingyao Xu
Peking University, P.R.China
Email: [email protected]
Guiying Yan
ChineseAcademyofMathematicsandSystem
Science, P.R.China
Email: [email protected]
Y. Zhang
Department of Computer Science
Georgia State University, Atlanta, USA
Famous Words:
If you want to know everything all at once, you will know nothing.
By Ivan Pavlov,a former Soviet physiologist, psychologist.
Math.Combin.Book Ser. Vol.4(2019), 1-18
Graphs, Networks and Natural Reality
– from Intuitive Abstracting to Theory
Linfan MAO
1.ChineseAcademyofMathematicsandSystemScience, Beijing100190, P.R.China
2.AcademyofMathematicalCombinatorics&Applications(AMCA),Colorado,USA
E-mail: [email protected]
Abstract: In theview of modern science, a matter is nothingelse but a complex network
→−G, i.e., the reality of matter is characterized by complex network. However, there are no
suchamathematicaltheoryoncomplexnetworkunlesslocalandstatisticalresults. Couldwe
establishsuchamathematicsoncomplexnetwork? Theanswerisaffirmative,i.e.,mathemat-
icalcombinatoricsormathematicsovertopologicalgraphs. Then,whatisagraph? Howdoes
it appears in the universe? And what is its role for understanding of the reality of matters?
Themainpurposeofthispaperistosurveytheprogressing processandexplainsthenotion
fromgraphstocomplexnetworkandthen,abstractsmathematicalelementsforunderstand-
ing reality of matters. For example, L.Euler’s solving on the problem of K¨onigsberg seven
bridgesresultedingraphtheoryandembeddinggraphsincompactn-manifold,particularly,
compact 2-manifold or surface with combinatorial maps and then, complex networks with
reality of matters. We introduce 2 kinds of mathematical elements respectively on living
body or non-living body for self-adaptive systems in the universe, i.e., continuity flow and
harmonicflow→−GL whichareessentiallyelementsinBanachspaceovergraphswithoperator
actions on ends of edges in graph →−G. We explain how to establish mathematics on the 2
kinds of elements, i.e., vectors underling a combinatorial structure →−G by generalize a few
well-known theorems on Banach or Hilbert space and contribute mathematics on complex
networks. Allof theseimply thatgraphsexpandthemathematical field,establish thefoun-
dation on holding on the nature and networks are closer more to the real but without a
systematic theory. However, its generalization enables one to establish mathematics over
graphs, i.e., mathematical combinatorics on reality of matters in theuniverse.
Key Words: Graph, 2-cell embedding of graph, combinatorial map, complex network,
reality, mathematical element, Smarandache multispace, mathematical combinatorics.
AMS(2010): 00A69,05C21,05C25,05C30 05C82, 15A03,57M20
§1. Introduction
What is the role of mathematics to natural reality? Certainly, as the science of quantity,
mathematics is the main tool for humans understanding matters, both for the macro and the
micro in the universe. Generally, it builds a model and characterizes the behavior of a matter
1ReceivedAugust25,2019,Accepted November20,2019.
2 LinfanMAO
for holding on reality and then, establishes a theory, such as those shown in Fig.1.
(cid:31) (cid:28) (cid:31) (cid:28)(cid:31) (cid:28)(cid:31) (cid:28)
Observation Induction Hypothesis
&Experiment - &Deduction - &Testing - Theory
(cid:30) (cid:29) (cid:30) (cid:29)(cid:30) (cid:29)(cid:30) (cid:29)
Fig.1
This scientific method on matters in the universe is completely reflected in the solving
process of L.Euler on the problem of K¨onigsberg seven bridges. Geographically, the city of
K¨onigsberg is located on both sides of Pregel River, including two large islands which were
connected to each other and the mainland by seven bridges, such as those shown in Fig.2.
The residents of K¨onigsberg usually wished to pass through each bridge once without repeat,
initialing at point of the mainland or islands.
Fig.2
However, no one traveled in such a way once. Then, a resident should how to travel for such a
walk? L.Euler solvedthis problem,and answeredit had no solution in 1736. How did he do it?
Let A,B,C,D be the two sides and islands. Then, he abstracted this problem on (a) equivalent
to finding a traveling passing through each lines on (b) without repeating.
(a) (b)
Fig.3
Clearly, such a traveling must be with the same in and out times at each point A,B,C or D.
But, (b) is not fitted with such conditions. So, there are no such a traveling in the problem on
Graphs,NetworksandNaturalReality–fromIntuitiveAbstractingtoTheory 3
K¨onigsberg seven bridges.
Euler’s solving method on the problem of K¨onigsberg seven bridges finally resulted graph
theory into beings today. A graph G is an ordered 3-tuple (V,E;I), where V,E are finite sets,
V = and I : E V V. Call V the vertex set and E the edge set of G, denoted by V(G)
6 ∅ → ×
and E(G), respectively. For example, two graphs K(3,3) and K are shown in Fig.4.
5
r
r r r
r r
r r
r r r
K
K(3,3) 5
Fig.4
Usually,if (u,v)=(v,u)for u,v V(G), then Gis calleda graph. Otherwise,it is called
∀ ∈
a directed graph with an orientation u v on each edge (u,v), denoted by −→G.
→
Let G = (V ,E ,I ), G = (V ,E ,I ) be 2 graphs. If there exists a 1 1 mapping
1 1 1 1 2 2 2 2
−
φ : V V and φ : E E such that φI (e) = I φ(e) for e E with the convention that
1 2 1 2 1 2 1
→ → ∀ ∈
φ(u,v) = (φ(u),φ(v)), then we say that G is isomorphic to G , denoted by G = G and φ
1 2 1 ∼ 2
an isomorphism between G andG . Clearly, allautomorphisms φ:V(G) V(G) of graphG
1 2
→
forma groupunder the compositionoperation,anddenotedbyAutGthe automorphismgroup
of graph G. A few automorphism groups of well-known graphs are listed in Table 1.
G AutG order
P Z 2
n 2
C D 2n
n n
K S n!
n n
K (m=n) S S m!n!
m,n m n
6 ×
K S [S ] 2n!2
n,n 2 n
Table 1
Certainly, an edge e = uv E(G) can be divided into two semi-arcs e ,e such as those
u v
∈
shown in Fig.5.
e Divided into eu ev
u u - u - (cid:27) u
u v u v
Fig.5
Similarly, two semi-arcse ,f are calledv-incident or e-incidentif u=v or e=f. Denote
u v
allsemi-arcsofagraphGbyX (G). A1 1mappingξ onX (G)suchthat e ,f X (G),
1 1 u v 1
2 − 2 ∀ ∈ 2
ξ(e ) and ξ(f ) are v incident or e incident if e and f are v incident or e incident, is
u v u v
− − − −
4 LinfanMAO
calledasemi-arcautomorphismofthegraphG. Clearly,allsemi-arcautomorphismsofagraph
also form a group, denoted by Aut G.
1
2
Certainly, graph theory studies properties of graphs. A property is nothing else but a
family of graph, i.e., P = G ,G , ,G , but closed under isomorphisms φ of graphs,
1 2 n
{ ··· ···}
i.e., Gφ P if G P. For example, hamiltonian graphs, Euler graphs and also interesting
∈ ∈
parameters, such as those of connectivity, independent number, covering number, girth, level
number, of a graph.
···
Themainpurposeofthispaperistosurveytheprogressingprocessandexplainsthenotion
from graphsto complex networkand then, abstracts mathematical elements for understanding
reality of matters. For example, L.Euler’s solving on the problem of K¨onigsberg seven bridges
resulted in graph theory and embedding graphs in compact n-manifold, particularly, compact
2-manifold or surface with combinatorial maps and then, complex networks with reality of
matters. We introduce 2 kinds of mathematical elements respectively on living or non-living
body in the universe, i.e., continuity and harmonic flows −→GL which are essentially elements
in Banach space over graphs with operator actions on ends of edges in graph −→G. We explain
howtoestablishmathematicsonthe2kindsofelements,i.e.,vectorsunderlingacombinatorial
structure−→G bygeneralizeafewwell-knowntheoremsonBanachorHilbertspaceandcontribute
a mathematics on complex networks.
For terminologiesand notations not mentionedhere, we follow references [1],[2]and[4] for
graphs, [3] for complex network, [6] for automorphisms of graph, [24] for algebraic topology,
[25] for elementary particles and [6],[26] for Smarandache systems and multispaces.
§2. Embedding Graphs on Surfaces
2.1 Surface
A surface is a 2-dimensionalcompact manifold without boundary. For example, a few surfaces
are shown in Fig.6.
Sphere Torus Klein bottle
Fig.6
Clearly,theintuitionimaginationisdifficultfordeterminingsurfaceofhighergenus. How-
ever, T.Rad´o showed the following result, which is the fundamental of combinatorialtopology,
Graphs,NetworksandNaturalReality–fromIntuitiveAbstractingtoTheory 5
ro topological graphs on surfaces.
Theorem2.1(T.Rad´o1925,[24]) ForanycompactsurfaceS,thereexistatriangulation T ,i
i
{ ≥
1 on S.
}
T.Rado´’sresultontriangulationofsurfaceenablesonetopresentasurfacebylistingevery
trianglewitheachsidealabelandadirection,i.e.,thepolygonrepresentation. Then,thesurface
is assembled by identifying the two sides with the same label and direction. This way results
in a polygon representation on a surface finally. For examples, the polygon representations on
surfaces in Fig.6 are shown in Fig.7.
b b
- A - -
6a
a
a 6 6a a 6 a
(cid:28)
(cid:28) ?
B (cid:28)(cid:28)
b - b -
Sphere Torus Klein bottle
Fig.7
We know the classification theorem of surfaces following.
Theorem2.2([24]) Anyconnectedcompact surfaceS iseitherhomeomorphic toasphere, orto
aconnectedsumof tori, ortoaconnectedsumof projective planes, i.e., itssurface presentation
is elementary equivalent to one of the standard surface presentations following:
S
(1) The sphere S2 = aaa 1 ;
−
|
(2) The connected sum of p tori
(cid:10) (cid:11)
p
T2#T2#···#T2 =* ai,bi,1≤i≤p | aibia−i 1b−i 1+;
i=1
p Y
| {z }
(3) The connected sum of q projective planes
q
P2#P2 #P2 = a ,1 i q a .
i i
··· * ≤ ≤ | +
q Yi=1
| {z }
A combinatorial proof on Theorem 2.2 can be found in [6]. By definition, the Euler
characteristic of is
S
χ( )= V( ) E( ) + F( ),
S | S |−| S | | S |
where V( ),E( ) and F( ) are respective the set of vertex set, edge set and face set of the
S S S
polygon representation of surface . Then, we know the next result.
S