scholarly journals Congruence Classes of Orientable $2$-Cell Embeddings of Bouquets of Circles and Dipoles

10.37236/313 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Yan-Quan Feng ◽  
Jin-Ho Kwak ◽  
Jin-Xin Zhou
Keyword(s):  
Orientable Surface ◽  
Simple Graph ◽  
Complete Graphs ◽  
Graph Automorphism ◽  
Wheel Graphs ◽  
Complete Bipartite Graphs ◽  
Multiple Edges ◽  
Orientable Surfaces ◽  
Surface Homeomorphism ◽  
Congruence Classes

Two $2$-cell embeddings $\imath : X \to S$ and $\jmath : X \to S$ of a connected graph $X$ into a closed orientable surface $S$ are congruent if there are an orientation-preserving surface homeomorphism $h : S \to S$ and a graph automorphism $\gamma$ of $X$ such that $\imath h =\gamma\jmath$. Mull et al. [Proc. Amer. Math. Soc. 103(1988) 321–330] developed an approach for enumerating the congruence classes of $2$-cell embeddings of a simple graph (without loops and multiple edges) into closed orientable surfaces and as an application, two formulae of such enumeration were given for complete graphs and wheel graphs. The approach was further developed by Mull [J. Graph Theory 30(1999) 77–90] to obtain a formula for enumerating the congruence classes of $2$-cell embeddings of complete bipartite graphs into closed orientable surfaces. By considering automorphisms of a graph as permutations on its dart set, in this paper Mull et al.'s approach is generalized to any graph with loops or multiple edges, and by using this method we enumerate the congruence classes of $2$-cell embeddings of a bouquet of circles and a dipole into closed orientable surfaces.

Neighbor Rupture Degree of Transformation Graphs Gxy−

2017 ◽  
Vol 28 (04) ◽  
pp. 335-355
Author(s):  
Goksen Bacak-Turan ◽  
Ekrem Oz
Keyword(s):  
Connected Graph ◽  
Bipartite Graphs ◽  
Connected Components ◽  
Complete Graphs ◽  
Maximum Order ◽  
Wheel Graphs ◽  
Complete Bipartite Graphs

A vulnerability parameter the neighbor rupture degree can be used to obtain the vulnerability of a spy network. The neighbor rupture degree of a noncomplete connected graph [Formula: see text] is defined to be [Formula: see text] where [Formula: see text] is any vertex subversion strategy of [Formula: see text], [Formula: see text] is the number of connected components in [Formula: see text], and [Formula: see text] is the maximum order of the components of [Formula: see text]. In this study, the neighbor rupture degree of transformation graphs [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] of path graphs, cycle graphs, wheel graphs, complete graphs and complete bipartite graphs are obtained.

scholarly journals Total Weight Choosability of Graphs

10.37236/599 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Jakub Przybyło ◽  
Mariusz Woźniak
Keyword(s):  
Main Tool ◽  
Bipartite Graphs ◽  
Simple Graph ◽  
Total Weight ◽  
Combinatorial Nullstellensatz ◽  
Complete Graphs ◽  
Complete Bipartite Graphs ◽  
Proper Vertex ◽  
Vertex Colouring

Suppose the edges and the vertices of a simple graph $G$ are assigned $k$-element lists of real weights. By choosing a representative of each list, we specify a vertex colouring, where for each vertex its colour is defined as the sum of the weights of its incident edges and the weight of the vertex itself. How long lists ensures a choice implying a proper vertex colouring for any graph? Is there any finite bound or maybe already lists of length two are sufficient? We prove that $2$-element lists are enough for trees, wheels, unicyclic and complete graphs, while the ones of length $3$ are sufficient for complete bipartite graphs. Our main tool is an algebraic theorem by Alon called Combinatorial Nullstellensatz.

scholarly journals Biembeddings of Metacyclic Groups and Triangulations of Orientable Surfaces by Complete Graphs

10.37236/2169 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Michael John Grannell ◽  
Martin Knor
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Complete Graph ◽  
Latin Squares ◽  
Orientable Surface ◽  
Complete Graphs ◽  
Metacyclic Group ◽  
Metacyclic Groups ◽  
Multiplicative Order ◽  
Orientable Surfaces

For each integer $n\ge 3$, $n\ne 4$, for each odd integer $m\ge 3$, and for any $\lambda\in\Bbb Z_n$ of (multiplicative) order $m'$ where $m'\mid m$, we construct a biembedding of Latin squares in which one of the squares is the Cayley table of the metacyclic group $\mathbb{Z}_m\ltimes_{\lambda}\mathbb{Z}_n$. This extends the spectrum of Latin squares known to be biembeddable.The best existing lower bounds for the number of triangular embeddings of a complete graph $K_z$ in an orientable surface are of the form $z^{z^2(a-o(1))}$ for suitable positive constants $a$ and for restricted infinite classes of $z$. Using embeddings of $\mathbb{Z}_3\ltimes_{\lambda}\mathbb{Z}_n$, we extend this lower bound to a substantially larger class of values of $z$.

scholarly journals On randomly 3-axial graphs

1982 ◽  
Vol 25 (2) ◽  
pp. 187-206
Author(s):  
Yousef Alavi ◽  
Sabra S. Anderson ◽  
Gary Chartrand ◽  
S.F. Kapoor
Keyword(s):  
Bipartite Graphs ◽  
Disjoint Paths ◽  
Complete Graphs ◽  
Initial Vertex ◽  
Complete Bipartite Graphs

A graph G, every vertex of which has degree at least three, is randomly 3-axial if for each vertex v of G, any ordered collection of three paths in G of length one with initial vertex v can be cyclically randomly extended to produce three internally disjoint paths which contain all the vertices of G. Randomly 3-axial graphs of order p > 4 are characterized for p ≢ 1 (mod 3), and are shown to be either complete graphs or certain regular complete bipartite graphs.

ON MOSTAR INDEX OF GRAPHS

2021 ◽  
Vol 10 (4) ◽  
pp. 2115-2129
Author(s):  
P. Kandan ◽  
S. Subramanian
Keyword(s):  
Connected Graph ◽  
Cartesian Product ◽  
Bipartite Graphs ◽  
Topological Indices ◽  
Great Success ◽  
Complete Graphs ◽  
Molecular Graphs ◽  
Graphs And Networks ◽  
Complete Bipartite Graphs ◽  
Simple Connected Graph

On the great success of bond-additive topological indices like Szeged, Padmakar-Ivan, Zagreb, and irregularity measures, yet another index, the Mostar index, has been introduced recently as a peripherality measure in molecular graphs and networks. For a connected graph G, the Mostar index is defined as $$M_{o}(G)=\displaystyle{\sum\limits_{e=gh\epsilon E(G)}}C(gh),$$ where $C(gh) \,=\,\left|n_{g}(e)-n_{h}(e)\right|$ be the contribution of edge $uv$ and $n_{g}(e)$ denotes the number of vertices of $G$ lying closer to vertex $g$ than to vertex $h$ ($n_{h}(e)$ define similarly). In this paper, we prove a general form of the results obtained by $Do\check{s}li\acute{c}$ et al.\cite{18} for compute the Mostar index to the Cartesian product of two simple connected graph. Using this result, we have derived the Cartesian product of paths, cycles, complete bipartite graphs, complete graphs and to some molecular graphs.

Graph-Theoretical Indices based on Simple, General and Complete Graphs

2013 ◽  
pp. 11-26
Author(s):  
Lionello Pogliani
Keyword(s):  
Complete Graphs ◽  
Molecular Connectivity ◽  
Theoretical Concepts ◽  
General Chemical ◽  
Starting Point ◽  
Wide Range ◽  
Connectivity Indices ◽  
Core Electrons ◽  
Multiple Edges ◽  
Electrotopological State

Valence molecular connectivity indices are based on the concept of valence delta, d v, that can be derived from general chemical graphs or chemical pseudographs. A general graph or pseudograph has multiple edges and loops and can be used to encode, through the valence delta, chemical entities. Two graph-theoretical concepts derived from chemical pseudographs are the intrinsic (I) and the electrotopological state (E) values, which are the used to define the valence delta of the pseudoconnectivity indices, ?I,S. Complete graphs encode, through a new valence delta, the core electrons of any atoms in a molecule. The connectivity indices, either valence connectivity or pseudoconnectivity, are the starting point to develop the dual connectivity indices. The dual indices show that not only can they assume negative values but also cover a wide range of numerical values. The central parameter of the molecular connectivity theory, the valence delta, defines a completely new set of connectivity indices, which can be distinguished by their configuration and advantageously used to model different properties and activities of compounds.

Some properties about the zero-divisor graphs of quasi-ordered sets

2019 ◽  
Vol 19 (04) ◽  
pp. 2050074
Author(s):  
Junye Ma ◽  
Qingguo Li ◽  
Hui Li
Keyword(s):  
Zero Divisor ◽  
Ordered Set ◽  
Line Graph ◽  
Prime Ideals ◽  
Complete Graphs ◽  
Ordered Sets ◽  
Graph Theoretic ◽  
Zero Divisor Graph ◽  
Complete Bipartite Graphs ◽  
Minimal Prime

In this paper, we study some graph-theoretic properties about the zero-divisor graph [Formula: see text] of a finite quasi-ordered set [Formula: see text] with a least element 0 and its line graph [Formula: see text]. First, we offer a method to find all the minimal prime ideals of a quasi-ordered set. Especially, this method is applicable for a partially ordered set. Then, we completely characterize the diameter and girth of [Formula: see text] by the minimal prime ideals of [Formula: see text]. Besides, we perfectly classify all finite quasi-ordered sets whose zero-divisor graphs are complete graphs, star graphs, complete bipartite graphs, complete [Formula: see text]-partite graphs. We also investigate the planarity of [Formula: see text]. Finally, we obtain the characterization for the line graph [Formula: see text] in terms of its diameter, girth and planarity.

Total Edge Irregularity Strength of Complete Graphs and Complete Bipartite Graphs

2007 ◽  
Vol 28 ◽  
pp. 281-285 ◽  
Author(s):  
Stanislav Jendrol' ◽  
Jozef Miškuf ◽  
Roman Soták
Keyword(s):  
Bipartite Graphs ◽  
Complete Graphs ◽  
Complete Bipartite Graphs ◽  
Irregularity Strength ◽  
Total Edge Irregularity Strength

On the homeotopy group of a non-orientable surface

1972 ◽  
Vol 71 (3) ◽  
pp. 437-448 ◽  
Author(s):  
Joan S. Birman ◽  
D. R. J. Chillingworth
Keyword(s):  
Projective Planes ◽  
Double Cover ◽  
Orientable Surface ◽  
Factor Group ◽  
Simple Type ◽  
Defining Relations ◽  
Entire Structure ◽  
Finite Set ◽  
Orientable Surfaces ◽  
Made In

Let X be a closed, compact connected 2-manifold (a surface), which we will denote by O or N if we wish to stress that X is orientable or non-orientable. Let G(X) denote the group of all homeomorphisms X → X, D(X) the normal subgroup of homeomorphisms isotopic to the identity, and H(X) the factor group G(X)/D(X), i.e. the homeotopy group of X. The problem of determining generators for H(O) was considered by Lickorish in (7, 8), and the second of these papers specifies a finite set of generators of a particularly simple type. In (10) and (11) Lickorish considered the analogous problem for non-orientable surfaces, and, using Lickorish's partial results, Chilling-worth (4) determined a finite set of generators for H(N). While the generators obtained for H(O) and H(N) were strikingly similar, it was noteworthy that the techniques used in the two cases were different, and in particular that little use was made in the non-orientable case of the earlier results obtained on the orientable case. The purpose of this note is to show that the results of Lickorish and Chillingworth on non-orientable surfaces follow rather easily from the work in (7, 8) by an application of some ideas from the theory of covering spaces (2). Moreover, while Lickorish and Chillingworth sought only to find generators, we are able to show (Theorem 1) how in fact the entire structure of the group H(N) is determined by H(O), where O is an orientable double cover of N. Finally, we are able to determine defining relations for H(N) for the case where N is the connected sum of 3 projective planes (Theorem 3).

scholarly journals The First Integral Cohomology of Pure Mapping Class Groups

2020 ◽  
Author(s):  
Javier Aramayona ◽  
Priyam Patel ◽  
Nicholas G Vlamis
Keyword(s):  
Cohomology Group ◽  
Class Group ◽  
Mapping Class Groups ◽  
First Integral ◽  
Orientable Surface ◽  
Mapping Class ◽  
Class Groups ◽  
Simplicial Homology ◽  
Integral Cohomology ◽  
Orientable Surfaces

Abstract It is a classical result that pure mapping class groups of connected, orientable surfaces of finite type and genus at least 3 are perfect. In stark contrast, we construct nontrivial homomorphisms from infinite-genus mapping class groups to the integers. Moreover, we compute the first integral cohomology group associated to the pure mapping class group of any connected orientable surface of genus at least 2 in terms of the surface’s simplicial homology. In order to do this, we show that pure mapping class groups of infinite-genus surfaces split as a semi-direct product.

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