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Counting structures in the Möbius ladder

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*...

##
###
Some Topological Invariants of Generalized Möbius Ladder
[article]

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. ...

##
###
Complexity continuum within Ising formulation of NP problems
[article]

2020
*
arXiv
*
pre-print

A compelling approach for distinguishing easy and hard instances within

arXiv:2008.00466v1
fatcat:fezdv73wzbaafk4gy3zlyfr3dq
*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. ...##
###
The M-Polynomial and Topological Indices of Generalized Möbius Ladder and Its Line Graph
[article]

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 . ...

##
###
On the Edge Metric Dimension of Different Families of Möbius Networks

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). ...

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

2004
*
Lecture Notes in Computer Science
*

We illustrate this by deriving recurrence relations

doi:10.1007/978-3-540-30551-4_45
fatcat:6xcw6m3y65fa5nx2rl5jdel55q
*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). ...##
###
Genus distributions of graphs under self-edge-amalgamations

2012
*
Ars Mathematica Contemporanea
*

prism graph, as well as graph families like circular and

doi:10.26493/1855-3974.166.63e
fatcat:b2dmjt3wcfdrvmpvu5sbsc4aye
*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 ...##
###
A classification of cubic bicirculants

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. ...

##
###
Interacting quantum walk on a graph
[article]

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*...

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

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. ...

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

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) = ...

##
###
Complexity of some special named graphs with double edges

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 ...

##
###
Topological elasticity of non-orientable ribbons
[article]

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. ...

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

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). ...

##
###
Embedding Quartic Eulerian Digraphs on the Plane
[article]

2017
*
arXiv
*
pre-print

Minimal obstructions for embedding 4-regular Eulerian digraphs on

arXiv:1706.02896v1
fatcat:xwe6l42x3jaztf35galjc3sjry
*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. ...
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