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Counting structures in the Möbius ladder

John P. McSorley
1998 Discrete Mathematics  
The Möbius ladder, M n , is a simple cubic graph on 2n vertices. We present a technique which enables us to count exactly many different structures of M n , and somewhat unifies counting in M n .  ...  Introduction The Möbius ladder, M n , is a simple cubic graph on 2n vertices. It is shown in Fig 1a, and the representation of it used in this paper appears in Fig. 1b .  ...  Even though some of these structures have been counted before, and thus some counts appear in the literature, it seems that this technique is new, and somewhat unifies the approach to counting structures  ... 
doi:10.1016/s0012-365x(97)00086-1 fatcat:py663zdborfgnli2t2sul23hle

Some Topological Invariants of Generalized Möbius Ladder [article]

Numan Amin, Abdul Rauf Nizami, Muhammad Idrees
2017 arXiv   pre-print
In this article we give the general closed form of the Hosoya polynomial of the generalized Möbius ladder M(m,n) for arbitrary m and for n=3.  ...  The Hosoya polynomial of a graph G was introduced by H. Hosoya in 1988 as a counting polynomial, which actually counts the number of distances of paths of different lengths in G.  ...  What we receive is the generalized Möbius ladder M m,n . You can see M 7,3 in the following figure.  ... 
arXiv:1708.09260v1 fatcat:mjmu2oraabe35oxx425oioziiu

Complexity continuum within Ising formulation of NP problems [article]

Kirill P. Kalinin, Natalia G. Berloff
2020 arXiv   pre-print
A compelling approach for distinguishing easy and hard instances within the same NP-hard class of problems can be a starting point in developing a standardised procedure for the performance evaluation  ...  The minimisation of the Ising Hamiltonian is known to be NP-hard problem for certain interaction matrix classes, yet not all problem instances are equivalently hard to optimise.  ...  For this percentage of rearranged edges, the still recognisable original four-band structure of the Mobius ladder graph has equivalent complexity of random 3-regular graphs.  ... 
arXiv:2008.00466v1 fatcat:fezdv73wzbaafk4gy3zlyfr3dq

The M-Polynomial and Topological Indices of Generalized Möbius Ladder and Its Line Graph [article]

Abdul Rauf Nizami, Muhammad Idrees, Numan Amin
2017 arXiv   pre-print
In this paper we give general closed forms of the M-polynomial of the generalized Möbius ladder and its line graph.  ...  The M-polynomial was introduced by Deutsch and Klavžar in 2015 as a graph polynomial to provide an easy way to find closed formulas of degree-based topological indices, which are used to predict physical  ...  Möbius ladder M 7The generalized Möbius ladder M 7,3 Figure 5 :. 5 M m,n .  ... 
arXiv:1708.08207v1 fatcat:ny2vuizp5rghjnal5tdbovabuq

On the Edge Metric Dimension of Different Families of Möbius Networks

Bo Deng, Muhammad Faisal Nadeem, Muhammad Azeem
2021 Mathematical Problems in Engineering  
In this paper, we study the edge resolvability parameter for different families of Möbius ladder networks and we find the exact edge metric dimension of triangular, square, and hexagonal Möbius ladder  ...  The minimum cardinality of an edge metric generator for G is called the edge metric dimension, and it is denoted by dimeG.  ...  Acknowledgments is study was supported by the NSFQH 11(no. 2018-ZJ-925Q).  ... 
doi:10.1155/2021/6623208 doaj:3aa872250d354750b0c70aa24c03a6cf fatcat:ohfe3ycfefhx5d65k4nxxrdcuy

Counting Spanning Trees and Other Structures in Non-constant-jump Circulant Graphs [chapter]

Mordecai J. Golin, Yiu Cho Leung, Yajun Wang
2004 Lecture Notes in Computer Science  
We illustrate this by deriving recurrence relations counting all of the structures listed above for various circulant graphs.  ...  For the non-constant-jump case, i.e., where some jump sizes can be functions of the graph size, only a few special cases such as the Möbius ladder had been studied but no general results were known.  ...  Since the size of matrix A is |P| = B(p(s+2s max )+s) where B(m) is the Bell number 2 of order m the order of the recurrence is at most B(p(s + 2s max ) + s).  ... 
doi:10.1007/978-3-540-30551-4_45 fatcat:6xcw6m3y65fa5nx2rl5jdel55q

Genus distributions of graphs under self-edge-amalgamations

Mehvish I. Poshni, Imran F. Khan, Jonathan L. Gross
2012 Ars Mathematica Contemporanea  
prism graph, as well as graph families like circular and Möbius ladders with previously known solutions to the genus distribution problem.  ...  We investigate the well-known problem of counting graph imbeddings on all oriented surfaces with a focus on graphs that are obtained by pasting together two root-edges of another base graph.  ...  Application: Revisiting circular ladders and Möbius ladders The genus distributions of circular ladders and Möbius ladders were first derived by [19] . §4 of [22] shows how calculation of the double-root  ... 
doi:10.26493/1855-3974.166.63e fatcat:b2dmjt3wcfdrvmpvu5sbsc4aye

A classification of cubic bicirculants

Tomaž Pisanski
2007 Discrete Mathematics  
The well-known Petersen graph G(5, 2) admits a semi-regular automorphism acting on the vertex set with two orbits of equal size. This makes it a bicirculant.  ...  Some basic properties of trivalent bicirculants are explored and the connection to combinatorial and geometric configurations are studied. Some analogues of the polycirculant conjecture are mentioned.  ...  Research was supported in part by a grant from Ministrstvo za šolstvo, znanost in šport Republike Slovenije. Part of the research was conducted while the author was Neil R.  ... 
doi:10.1016/j.disc.2005.09.053 fatcat:3yl4dtypqnevbfvyepfnp6epkq

Interacting quantum walk on a graph [article]

Alberto D. Verga
2018 arXiv   pre-print
The quantum state is build sequentially, applying a unitary transformation that couples neighboring spins and, at a node, the local spin with the particle.  ...  We observe the relaxation of the system towards a stationary paramagnetic or ferromagnetic state, and demonstrate that it is related to eigenvectors thermalization and random matrix statistics.  ...  This is visible in the case of the chain graph (central column in Fig. 4) , where the maximum entropy eigenvector shows a regular structure that contrasts with the extended eigenvectors of the ladder  ... 
arXiv:1805.02929v1 fatcat:7xwz2izi65dhridu5eaoo5lyuy

0, 1/2‐Cuts and the Linear Ordering Problem: Surfaces That Define Facets

Samuel Fiorini
2006 SIAM Journal on Discrete Mathematics  
In that sense, Möbius ladder inequalities have the same 'shape' as the projective plane.  ...  Our starting point is to observe that the natural derivation of the Möbius ladder inequalities as {0, 1 2 }-cuts produces triangulations of the Möbius band and of the corresponding (closed) surface, the  ...  Schulz for their early interest in the results of this article.  ... 
doi:10.1137/s0895480104440985 fatcat:lg3elyeogfh3xjrh4vvqdflvi4

Counting Structures in Grid Graphs, Cylinders and Tori Using Transfer Matrices: Survey and New Results

Mordecai J. Golin, Yiu-Cho Leung, Yajun Wang, Xuerong Yong
2005 Workshop on Algorithm Engineering and Experimentation  
In particular the problem of counting the number of structures in fixed height graphs, i.e., fixing m and letting n grow, has been, for different types of structures, attacked independently by many different  ...  In contrast there has been surprisingly little work done on counting structures in grid-cylinders (where the left and right, or top and bottom, boundaries of the grid are wrapped around and connected to  ...  Thus, the techniques in this paper easily permit counting all types of structures on the Mobius ladder (which was recently done in a different way by [11] ). and A(S, T, m) = A(S, T C, m) b(S, T, m) =  ... 
dblp:conf/alenex/GolinLWY05 fatcat:t4svosjsibg5tojpf5ja25vf6a

Complexity of some special named graphs with double edges

Salama N. Daoud, Kamel A. Mohamed
2014 Journal of Taibah University for Science  
In this paper we derive simple formulas of the complexity, number of spanning trees, of Some Special named Graphs with double edges such as Fan, Wheel and Mobius ladder, using linear algebra, Chebyshev  ...  In mathematics, one always tries to get new structures from given ones. This also applies to the realm of graphs, where one can generate many new graphs from a given set of graphs.  ...  Acknowledgements The authors are deeply indebted and thankful to the deanship of the scientific research for his helpful and for distinct team of employees at Taibah university, Al-Madinah Al-Munawarah  ... 
doi:10.1016/j.jtusci.2013.08.002 fatcat:4x5cgj3w6bb7fisyc2ylxbdzyi

Topological elasticity of non-orientable ribbons [article]

Denis Bartolo, David Carpentier
2019 arXiv   pre-print
In this article, we unravel an intimate relationship between two seemingly unrelated concepts: elasticity, that defines the local relations between stress and strain of deformable bodies, and topology  ...  We establish a quantitative connection between the modes found at the interface between inequivalent topological insulators and solitonic bending excitations that freely propagate through the bulk non-orientable  ...  By contrast, the buckled Möbius ladder sketched in Fig. 6c is a closed isostatic system with a vanishing Maxwell-Caladine index.  ... 
arXiv:1910.06179v1 fatcat:lsr6pata7vc5hgyizhx4ni3oh4

On complexity and Jacobian of cone over a graph [article]

L. A. Grunwald, I.A. Mednykh
2020 arXiv   pre-print
In this paper, we study two important invariants of such a cone. Namely, complexity (the number of spanning trees) and the Jacobian of a graph.  ...  We prove that complexity of graph G coincides the number of rooted spanning forests in graph G and the Jacobian of G is isomorphic to cokernel of the operator I+L(G), where L(G) is Laplacian of G and I  ...  The cone over the Möbius ladder M (n). Recall that the Möbius ladder M(n) is circulant graph C 2n (1, n).  ... 
arXiv:2004.07452v1 fatcat:qwddgrouevdgpjvyphlayv4iva

Embedding Quartic Eulerian Digraphs on the Plane [article]

Dan Archdeacon, Paul Bonnington, Bojan Mohar
2017 arXiv   pre-print
Minimal obstructions for embedding 4-regular Eulerian digraphs on the plane are considered in relation to the partial order defined by the cycle removal operation.  ...  The results are based on concepts first discovered in Auckland in December, 2004.  ...  This gives a digraph − → M + n called the Möbius ladder since it can be obtained from the usual (undirected) Möbius ladder with n spokes by replacing every other rim edge with a digon.  ... 
arXiv:1706.02896v1 fatcat:xwe6l42x3jaztf35galjc3sjry
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