Mathematical combinatorics : Vol. 4, 2018 / : international book series.

2018·1.7 MB·English

by  Mao, Linfan

ISBN 978-1-59973-588-7
VOLUME 4, 2018
MATHEMATICAL COMBINATORICS
(INTERNATIONAL BOOK SERIES)
Edited By Linfan MAO
THE MADIS OF CHINESE ACADEMY OF SCIENCES AND
ACADEMY OF MATHEMATICAL COMBINATORICS & APPLICATIONS, USA
December, 2018
Vol.4, 2018 ISBN 978-1-59973-588-7
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, 2018
Aims and Scope: The Mathematical Combinatorics (International Book Series) is
a fully refereed international book series with ISBN number on each issue, sponsored by the
MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 110-160
pages approx. per volume, which publishes original research papers and survey articles in all
aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics,
non-euclidean geometry and topology and their applications to other sciences. Topics in detail
to be covered are:
Smarandache multi-spaces with applications to other sciences, such as those of algebraic
multi-systems, multi-metric spaces, , etc.. Smarandache geometries;
···
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;
Applications of Smarandache multi-spaces to theoretical physics; Applications of Combi-
natorics to mathematics and theoretical physics; Mathematical theory on gravitational fields;
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combinatorics.
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Email: [email protected]
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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:
The mathematician lives long and lives young; the wings of his soul do not
early drop off, nor do its pores become clogged with the earthy particles blown from
the dusty highways of vulgar life.
By James Joseph Sylvester, a British mathematician.
Math.Combin.Book Ser. Vol.4(2018), 1-17
A Combinatorial Approach
For the Spanning Tree Entropy in Complex Network
E.M.Badr
(ScientificComputingDepartment,FacultyofComputersandInformatics,BenhaUniversity,Benha,Egypt)
B.Mohamed
(DepartmentofMathematicsandComputerScience,FacultyofScience,MenoufiaUniversity,Menoufia,Egypt)
E-mail: [email protected],[email protected]
Abstract: The goal of this paper is to propose the combinatorial method to facilitate
the calculation of the number of spanning trees for complex networks. In particular, we
derivetheexplicitformulasforthetriangularsnake,doubletriangularsnake,fourtriangular
snake, the total graph of path, the generalized friendship graphs and the subdivision of
doubletriangular snake. Finally, we calculate theirspanning trees entropy and we compare
it between them.
Key Words: Entropy,cyclic snakes, total graph, numberof spanning trees.
AMS(2010): 05C05, 05C30.
§1. Introduction
In real life, most of the systems are represented by graphs, such that the nodes denote the
basic constituents of the system and edges describe their interaction. The Internet, electric,
bioinformatics, telephone calls, social networks and many other systems are now represented
by complex graphs [1].
There are many differenttypes ofnetworksand their classificationdepends on the proper-
ties such as nodes degrees, clustering coefficients, shortest paths. Another concern in studying
complex network is how to evaluate the robustness of a network and its ability to adapt to
changes [21]. The robustness of a network is correlated to its ability to deal with internal
feedbackswithinthe networkandtoavoidmalfunctioningwhena fractionofits constituentsis
damaged. We usethe entropyofspanningtreesorwhatis calledthe asymptoticcomplexity [4]
in order to quantify the robustness and to characterizethe structure. The number of spanning
trees in G, also called, the complexity of the graph is a well-studied quantity (for long time)
and appear in a number of applications. Most notable applicationfields are network reliability
[15, 16, 17], enumerating certain chemical isomers [18] and counting the number of Eulerian
circuits in a graph [19].
1ReceivedMarch7,2018,Accepted November15,2018.
2 E.M.BadrandB.Mohamed
A graph G has different subgraphs. In fact a graph having V(G) nodes has
| |
V(G)(V(G) 1)
2(| | | − | )
2
possible distinct subgraphs. Some of these subgraphs are trees and the others are not trees.
We are focused certain kinds of trees called spanning trees. The history of determining the
number of spanningtrees τ(G) of a graphG, dates backto the year1842in whichthe German
MathematicianGustavKirchhoff[2]introducedarelationbetweenthenumberofspanningtrees
of a graph G, and the determinant of a specific submatrix associated with G. This method is
infeasible for large graphs. For this reason scientists have developed techniques to get around
thedifficultiesandhavepaidmoreattentiontoderivingexplicitandsimpleformulasforspecial
classes, see [3 - 13].
Thebasiccombinatorialidea,Feussnersrecursiveformula[20],forcountingτ(G)inagraph
G is quite intuitive. For an undirected simple graph G, let e be any edge of G. All spanning
trees in G can be separatedinto two parts: one part contains all spanning trees without e as a
tree edge; the other part contains all spanning trees with e as a tree edge. The first part has
the same number of spanning trees as graph G e , but leaving all other edges and vertices
−
as they are. The second part has the same number of spanning trees as graph G e, where
⊙
G eis the graph (not a subgraph) obtained from G by contracting the edge e = u,v until
⊙ { }
the two vertices u and v coincide. Call this new vertex uv. Both G e and G e have fewer
− ⊙
edges, than G. So the number of spanning trees in G can be counted recursively in this way.
In this paper, we propose the combinatorialmethod to facilitate the calculationof the number
of spanning trees for complex networks. In particular, we derive the explicit formulas for
the triangular snake (∆ snake), double triangular snake (2∆ snake), four triangular
k k
− −
snake(4∆ snake),thetotalgraphofpathP (T(P )), thegraphnC 2P ,thegeneralized
k n n 4 n
− ⊙
friendshipgraphskF andthesubdivisionofdoubletriangularsnake(S(2∆ snake)). Finally,
n n
−
we calculate their spanning trees entropy and we compare it between them.
§2. Preliminary Notes
The combinatorial method involves the operation of contraction of an edge. An edge e of a
graphG is saidto be contractedif it is deleted and its ends are identified. The resulting graph
is denoted by G e . Also we denote by G e the graphobtained from G by deleting the edge
• −
e.
Theorem 2.1([13-20]) Let G be a planar graph (multiple edges are allowed in here). Then for
any edge τ(G)=τ(G e)+τ(G e).
− •
Definition 2.2([22]) A triangular snake(∆ snake) is a connected graph in which all blocks
−
are triangles and the block-cut-point graph is a path, as shown in Figure 1.
Definition 2.3 For an integer number m, an m-triangular snake is a graph formed by m
triangular snakes having a common path. If m = 2 that graph is called the double triangular
ACombinatorialApproachfortheSpanningTreeEntropyinComplexNetwork 3
snake is denoted by 2∆ snake, as shown in Figure 1.
−
Definition 2.4 The friendship graph F is a collection of k-cycles (all of order n), meeting
n,k
at a common vertex, as shown in Figure 1.
Definition 2.5 The graph nC 2P is a connected graph obtained from n copies of C
m n m
⊙
( nC is a disconnected graph) and two paths where each path connects with one vertex u
m i
(i = 1,2, ,2n) of each copy of C . All the vertices u (i = 1,2, ,2n) are distinct as
m i
··· ···
shown in Figure 1.
r r r r r r r r
r r r r r r r r r r r
r r r
(a) ∆k-snake (b) 2∆3-sanke (c) T(P3)
r r r
r r r r r r r r r (cid:12)r(cid:12) r
r r
!!! r r r r r r r r r r (cid:12)r r r
!r
r r r r r r
r r r r r r r r
r r
(f) S(2∆ snake)
(d) F (e) 3C4 2P3 3−
3,4 ⊙
Figure 1 Triangular snake, double triangular snake, four triangular snake,
total graph of path, generalized friendship and subdivision of double triangular snake
Definition 2.6 The total graph of a graph G is the graph whose vertex set is V(G) E(G) and
∪
two vertices are adjacent whenever they are either adjacent or incident in G. The total graph
of G denoted by T(G).
§3. Main Results
Theorem 3.1 The number of spanning trees of triangular snake graph is
τ(∆ )=3n.
n
′
Proof Consider a triangular snake graph ∆ constructed from ∆ by deleting one edge.
n n
See Figure 2.
4 E.M.BadrandB.Mohamed
r r r r r r
r r r r r r r r r r
∆
n ∆
′n
Figure 2 Triangular snake graph (∆ )
n
We put
′ ′
∆ =τ(∆ ) and ∆ =τ(∆ ).
n n n n
It is clear that
′ ′ ′
∆ =2(∆ )+3(∆ ) and ∆ =2(∆ ) 3(∆ )
n n−1 n−1 n n−1 − n−1
′
with initial conditions ∆ =3,∆ =1 thus we have
1 n
∆ ∆
n n
=A ,
∆′  4∆′ 
n n
   
where,
2 3 ∆ ∆ ∆
A=2 3; ∆n′ =A∆n′−1=···=An−1∆1′,
− n n−1 1
       
we compute An 1as follows:
−
det(A λI )=λ2 λ 12=0, λ = 4 and λ =3,λ =λ .
2 1 2 1 2
− − − − 6
Then there is a matrix M is invertible such that A=MBM 1, where
−
λ 0
1
B =
0 λ 
2
 
and M is an invertible transformation matrix formed by eigenvectors
1 1 1 3
M = ; M 1 7 −7 ; An 1 =MBn 1M 1,
− − − −
 2 1 6 3 
− 3 7 7
   
where
( 4)n 1 0
Bn 1 = − −
−
 0 (3)n 1
−
 
ACombinatorialApproachfortheSpanningTreeEntropyinComplexNetwork 5
From which, we obtain
An−1 =−2∗((−−44)7)nn−−11 ++ 22∗∗7(33n)n−1 6−∗3(−∗(4−)7n4)−n1−+1 +3n3−7n1
7 7 7 7
 
and hence the result follows.
2
Theorem 3.2 The number of spanning trees of the double triangular snake is
τ(2∆ snake)=8n.
n
−
Proof Consider a double triangular snake graph 2∆ -snake constructed from 2∆ -snake
′n n
by deleting two edges. See Figure 3.
r r r r r r
L
L
r r r r r r Lr r r r
r r r r r r
2∆ -snake
n 2∆ -snake
′n
Figure 3 Triangular snake graph (∆ )
n
We put
′ ′
2∆ snake=τ(2∆ snake) and 2∆ snake=τ(2∆ snake).
n− n− 2− 2−
It is clear that
′
2∆ snake = 7(2∆ snake)+8(2∆ snake)
n− n−1− 2−
′ ′
2∆ snake = 2(2∆ snake) 8(2∆ snake)
2− n−1− − n−1−
′
with initial conditions 2∆ snake=8, 2∆ snake=1. Thus we have
1− 1−
2∆ snake 2∆ snake 7 8
n n 1
− =A − − , where A= ,
2∆′ snake  2∆′ snake  2 8
n− n− −
     
2∆ snake 2∆ snake 2∆ snake
2∆n′ −snake=A 2∆n−′ 1−snake =···=An−12∆1′ −snake.
n− n− 1−
     
We compute An 1 as follows:
−
det(A λI )=λ2 λ 72=0, λ = 9 and λ =8, λ =λ .
2 1 2 1 2
− − − − 6