Enumeration of the Number of Spanning Trees of the Globe Network and Its Subdivision

Enumeration of the Number of Spanning Trees of the
Globe Network and Its Subdivision
Basma Mohamed
Department of Computer Science, Faculty of Computers and Artificial Intelligence,
AlRyada University for Science and Technology, Sadat City, Egypt
Mohamed Amin
Mathematics and Computer Science Department, Faculty of Science, Menoufia University,
Shebin Elkom, 32511, Egypt
Abstract
The complexity of a graph can be determined using a network-theoretic method, which is
given. This strategy is based on the connection between graph theory and the theory of
determinants in linear algebra. In this article, we provide a unique algebraic method that
makes use of linear algebra to derive simple formulas for the complexity of diverse novel
networks. We can derive the explicit formulas for the globe network and its subdivision
using this method. In the final least, we also determine their spanning trees entropy and
compare it to each other.
Mathematics Subject Classification: Primary 05C05, Secondary 05C30
Keywords spanning trees, Laplacian matrix, entropy
1 Introduction
One of the well-researched topics in graph theory is determining the number of spanning
trees in a graph. We deal with simple and finite graphs G = (V, E), where V denotes the
vertex set and E denotes the edge set. A graph T is called a tree if it has not circuits so there
is exactly one path connecting each vertex to every other vertex in the tree. Spanning tree of
a graph G is a tree that includes every vertex in the graph. The number of spanning trees in
G, also called, the complexity of the graph G, indicated by (G)[1], can be found in many
applications.
The most important application areas are reliability of networks [2-4], recalling specific
chemical isomers [5], and determining the number of Eulerian circuits in a graph [1]. In
instance, counting spanning trees is an important phase in many computation, bounds, and
approximation algorithms for network dependability [6]. In a network that can be represented
by a graph, intercommunication between all nodes of the network necessitates the existence
of a spanning tree; hence increasing the number of spanning trees is a method of increasing
International Journal on Applications of Graph Theory in Wireless Ad hoc Networks and Sensor Networks (GRAPH-HOC) Vol.15, No.3, September 2023
DOI: 10.5121/jgraphoc.2023.15301 1

reliability.
A classic technique called the matrix tree theorem, also known as Kirchhoff''s matrix-tree
theorem [7] which states that the number of non-identical spanning trees of a graph G is
equal to any cofactor of its Laplacian matrix L=D-A, in which D is the degree matrix and A is
the adjacency matrix of the graph G.
Using Laplacian eigenvalues is a another way to count this number. The following formula
was created by Kelmans and Chelnoknov [8]:
( ) ∏ (1.1)
where p= are the eigenvalues of the Laplacian matrix L and G is a
connected graph with p vertices.The deletion-contraction method is one common way to find
the number of spanning tress (G). This method, which is reliable, enables counting the
number of spanning trees in a multi graph G. The fact that is used in this method.
τ(G)= τ(G-e)+ τ(G/e) (1.2)
where G-e denotes the graph obtained by deleting an arbitrary edge e, and G/e denotes the
graph obtained by contracting an arbitrary edge e [1, 9]. For more results, see [10- 26].
In this paper we contribute a new algebraic method which is based on Dodgson and Chio’s
method. It has an advantage that calculate determinants of n × n (n = 3) matrix, by reducing
determinants to 2 Order.
2.Dodgson and Chio’s Condensation Method: Chio’s condensation is a method for
analyzing an n×n determinant in terms of (n-1)×(n-1) determinants[27]:
A=| |
|
|
| | | |
| | | |
| |
| |
| | | | | |
|
|
(2.1)
In order to compute determinants of size n× n, Dodgson's condensation method first
expresses them in terms of determinants of size (n-1) × (n-1) before expressing the latter in
terms of determinants of size (n-2) × (n-2).
Another method, proposed by Armend [28] is based on Dodgson and Chio's method, but
differs from it in that it resolves the problem by computing four unique determinants of (n-1)
× (n-1) Order (which can be derived from determinants of n× n order; if the first row and the
first column, the first row and the last column, the last row and the first column, the last row
and the last column and elements that belong to only one of unique determinants are
removed, we should refer to them unique elements and one determinant of (n-2)×(n-2) order
which is formed from n× n order determinant with elements ai,j with i, j ≠1, n, on condition
that the determinant of (n-2)×(n-2)≠0.
Theorem 2.1 [28] States that every determinant of n× n (n > 2) order can be reduced into a
determinant of 2×2 order by computing 4 determinants of (n-1) × (n-1) order and one
determinant of (n-2) × (n-2) order, on condition that (n-2)×(n-2) order determinants to be
different from zero.
2

A method for computing the determinants of the n× n order using the following formula is
currently being presented:
A=| | | |
|
| | | |
| | | |
| | | (2.2)
The |B| is (n-2) ×(n-2) order determinant which is the interior determinant of determinant |A|
while |C|, |D|, |E| and |F| are unique determinants of (n-1)×(n-1)order, which can be formed
from n× n order determinant.
3. Main results
Theorem 3.1: The number of spanning trees of globe graph GLn is
(GLn) =(k+1)*2k
, k ≥ 2, where k is the number of blocks.
Proof:
Let n = k+3 and q = 2k+2 are the number of vertices and edges of GLn respectively.
( ) |
| |
|
According to Dodgson and Chio's method, we have
( ) |
| |
|
| |
|
| | | |
| | | |
|= (k+1)*2k
. | |≠0 such that:
| | | |= 2k
, | | |
|
|
|=2k-1
*(k+2)
| | |
|
|
|=2k
| | |
|
|
|=2k
| | |
|
|
|=2k+1
Therefore we get
3

( ) |
| |
|
| |
|
| | | |
| | | |
|= | ( ) | =(k+1)*2k
,
k ≥2
Theorem 3.2: The number of spanning trees of subdivision of globe graph S(GLn) is
(S(GLn)) =(k+1)*4k
, k ≥ 2, where k is the number of blocks.
Proof:
Let p=3k+5 and q=4k+4 are the number of vertices and edges of (S(GLn)) respectively.
According to Dodgson and Chio's method, we have
| |
|
| | | |
| | | |
|= (k+1)*4k
. | |≠0 such that:
𝜏(𝑆(𝐺𝐿𝑛))=
𝜏(𝑆(𝐺𝐿𝑛))
|B|= 𝑘
4

|C|=
| |
|
| | | |
| | | |
|= (k+1)*4k
, k ≥ 2.
4. Spanning Tree Entropy: The entropy of spanning trees of a network or the asymptotic
complexity is a quantitative measure of the number of spanning trees and it characterizes the
network structure. We use this entropy to quantify the robustness of networks. The most
robust network is the network that has the highest entropy. We can calculate its spanning tree
entropy which is a finite number and a very interesting quantity characterizing the network
structure, defined as in [29] as
Z(G)= ( )
( )
| ( )|
(4.1)
Corollary 4.1: The entropy of spanning trees of the globe graph GLn is
Z(GLn)=2
Proof: From the Theorem 3. 1 and equation (4.1) and |V(GLn) | = n = k +3 we obtain:
Z(GLn)= =
( )
√ =2.
Corollary 4.2: The entropy of spanning trees of the subdivision of globe graph S(GLn) is
Z (S(GLn))=2
Proof: From the Theorem 3. 2 and equation (a) and |V(GLn)| = n = 3k +5 we obtain:
𝑘
(𝑘 )
|D|=
𝑘
|E|=
𝑘
|F|=
𝑘
𝜏(𝑆(𝐺𝐿𝑛))
5

Z(GLn)= =
( ) ( )
√ = 2.
5. Conclusion
In this paper, we contributed a new algebraic method to derive simple formulas for the
complexity of some new networks using linear algebra. We applied this method to derive the
explicit formulas for the globe network and its subdivision.
References
[1] N.L. Biggs, Algebraic Graph Theory. 2nd Edn., Cambridge Univ. Press, Cambridge, 1993.
[2] W. Myrvold, K.H. Cheung, L.B. Page and J.E. Perry, Uniformly-most reliable networks do not
always exist, Networks, 21(1991), 417-419.
[3] S. Lata, S. Mehfuz, S. Urooj, A. Ali and N. Nasser,Disjoint spanning tree based reliability
evaluation of wireless sensor network, Sensors, 20(11), 2020.
[4] C. Y. Lam and W. H. Ip, An improved spanning tree approach for the reliability analysis of supply
chain collaborative network, Enterp. Inf. Syst., 6(4), 2012, 405-418.
[5]T.J.N. Brown, R.B. Mallion, P. Pollak and A. Roth, Some methods for counting the spanning trees
in labeled molecular graphs, examined in relation to certain fullerenes, Discrete
Appl. Math., 67, 1996, 51-66.
[6] C.J. Colbourn, The Combinatorics of Network Reliability, OUP,1978.
[7] G.G. Kirchhoff, Über die Auflösung der Gleichungen auf welche man bei der Untersucher der
linearen Verteilung galuanischer Strome gefhrt wird , Annalen der Physik, 148(12),1847,497-508.
[8] A.K. Kelmans, V.M. Chelnokov, A certain polynomial of a graph and graphs with an extremal
number of trees, J. Comb. Theory, B, 16(3),1974, 197-214.
[9] S. N. Daoud, The deletion-contraction method for counting the number of spanning trees of
graphs, Eur. Phys. J. Plus, 130, 2015,1-14.
[10] S. N. Daoud, Complexity of graphs generated by wheel graph and their asymptotic limits,
JOEMS, 25)4(, 2017, 424-433.
[11] S. N. Daoud, Number of spanning trees in different products of complete and complete tripartite
graphs, Ars Comb., 139, 2018, 85-103.
[12] S. N. Daoud, Number of Spanning trees of cartesian and composition products of graphs and
Chebyshev polynomials, IEEE Access, 7, 2019, 71142-71157.
[13] J. B. Liu, and S. N. Daoud, The complexity of some classes of pyramid graphs created from a
gear graph, Symmetry, 10 (12), 2018.
[14] S. N. Daoud and W. Saleh, Number of Spanning Trees of Some of Pyramid Graphs Generated by
a Wheel Graph, Math. Comb, 43, 2020.
[15] S. N. Daoud, Complexity of join and corona graphs and Chebyshev polynomials, Journal of
Taibah University for Science, 12(5), 2018, 557-572.
[16] D. Li, W. Chen, and W. Yan, Enumeration of spanning trees of complete multipartite graphs
containing a fixed spanning forest, J Graph Theory, 2023.
[17] M. N. Yanhaona, A. S. Nomaan, and M. S. Rahman, Efficiently Enumerating All Spanning
Trees of a Plane 3-Tree, International Conference on Algorithms and Complexity. Cham: Springer
International Publishing, 2023.
[18] N. Forsgren, Methods from Linear Algebra for the Enumeration of Spanning Trees, 2023.
[19] M. R. Deen, Enumeration of spanning trees in prisms of some graphs, Proyecciones
(Antofagasta) 42(2), 2023, 339-391.
[20] B. Mohamed, L. Mohaisen and M. Amin, Binary Equilibrium Optimization Algorithm for
Computing Connected Domination Metric Dimension Problem, Sci. Program, 2022.
[21] B. Mohamed and M. Amin, The Metric Dimension of Subdivisions of Lilly Graph, Tadpole
Graph and Special Trees," Appl. Comput. Math, 12(1), 2023, 9-14.
[22] B. Mohamed, L. Mohaisen and M. Amin, Computing Connected Resolvability of Graphs Using
Binary Enhanced Harris Hawks Optimization, Intell. Autom. Soft Comput, 36(2), 2023.
6

[23] B. Mohamed, Metric Dimension of Graphs and its Application to Robotic Navigation, IJCA,
184(15), 2022.
[24] B. Mohamed and M. Amin, Domination Number and Secure Resolving Sets in Cyclic Networks,
Appl. Comput. Math, 12(2), 2023, 42-45.
[25] B. Mohamed and M. Amin, A hybrid optimization algorithms for solving metric dimension
problem, (GRAPH-HOC),15(1),2023,1-10.
[26] B. Mohamed, A Comprehensive Survey on the Metric Dimension Problem of Graphs and Its
Types, Int. J. Theor. Appl. Math ,9(1),2023,1-5.
[27] Q. Gjonbalaj and A. Salihu, Computing the determinants by reducing the orders by
four, Appl.Math. E-Notes, 10, 2010, 151-158.
[28] A. Salihu, New method to calculate determinants of n× n (n≥ 3) matrix, by reducing
determinants to 2nd
order, Int. J. Algebra, 6(19), 2012, 913-917.
[29] R. Lyons, Asymptotic enumeration of spanning trees, Comb. Probab. Comput, 14(4),
2005, 491-522.
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