On the Maximum Order of the Automorphism Group of a Planar Triply Connected Graph

1966 ◽  
Vol 14 (4) ◽  
pp. 729-738 ◽  
Author(s):  
Louis Weinberg
Keyword(s):  
Automorphism Group ◽  
Connected Graph ◽  
Maximum Order

The Maximum Order of the Group of a Tournament

1967 ◽  
Vol 10 (4) ◽  
pp. 503-505 ◽  
Author(s):  
John D. Dixon
Keyword(s):  
Automorphism Group ◽  
Maximum Order ◽  
Image Position

To each tournament Tn with n nodes n there corresponds the automorphism group G(Tn) consisting n of all dominance preserving permutations of the set of nodes. In a recent paper [3], Myron Goldberg and J. W. Moon consider the maximum order g(n) which the group of a tournament with n nodes may have. Among other results they prove that12

scholarly journals Discrete Schrödinger operators on a graph

1992 ◽  
Vol 125 ◽  
pp. 141-150 ◽  
Author(s):  
Polly Wee Sy ◽  
Toshikazu Sunada
Keyword(s):  
Riemannian Manifold ◽  
Automorphism Group ◽  
Connected Graph ◽  
Spectral Properties ◽  
Orbit Space ◽  
Finite Graph ◽  
Discrete Analogue ◽  
Discrete Laplacian ◽  
Locally Finite ◽  
Discrete Schrödinger Operators

In this paper, we study some spectral properties of the discrete Schrödinger operator = Δ + q defined on a locally finite connected graph with an automorphism group whose orbit space is a finite graph.The discrete Laplacian and its generalization have been explored from many different viewpoints (for instance, see [2] [4]). Our paper discusses the discrete analogue of the results on the bottom of the spectrum established by T. Kobayashi, K. Ono and T. Sunada [3] in the Riemannian-manifold-setting.

scholarly journals On the maximum orders of an induced forest, an induced tree, and a stable set

2014 ◽  
Vol 24 (2) ◽  
pp. 199-215
Author(s):  
Alain Hertz ◽  
Odile Marcotte ◽  
David Schindl
Keyword(s):  
Connected Graph ◽  
Stable Set ◽  
Stability Number ◽  
Maximum Order ◽  
The Stability ◽  
Special Case

Let G be a connected graph, n the order of G, and f (resp. t) the maximum order of an induced forest (resp. tree) in G. We show that f - t is at most n - ?2?n-1?. In the special case where n is of the form a2 + 1 for some even integer a ? 4, f - t is at most n - ?2?n-1?-1. We also prove that these bounds are tight. In addition, letting ? denote the stability number of G, we show that ? - t is at most n + 1- ?2?2n? this bound is also tight.

Upper triangle free detour number of a graph

2021 ◽  
pp. 2150094
Author(s):  
S. Sethu Ramalingam ◽  
S. Athisayanathan
Keyword(s):  
Free Path ◽  
Connected Graph ◽  
Proper Subset ◽  
Maximum Order ◽  
Minimum Order ◽  
Geodetic Number ◽  
Number Formula ◽  
Positive Integers ◽  
Distance Formula

For any two vertices [Formula: see text] and [Formula: see text] in a connected graph [Formula: see text], the [Formula: see text] path [Formula: see text] is called a [Formula: see text] triangle free path if no three vertices of [Formula: see text] induce a triangle. The triangle free detour distance [Formula: see text] is the length of a longest [Formula: see text] triangle free path in [Formula: see text]. A [Formula: see text] path of length [Formula: see text] is called a [Formula: see text] triangle free detour. A set [Formula: see text] is called a triangle free detour set of [Formula: see text] if every vertex of [Formula: see text] lies on a [Formula: see text] triangle free detour joining a pair of vertices of [Formula: see text]. The triangle free detour number [Formula: see text] of [Formula: see text] is the minimum order of its triangle free detour sets and any triangle free detour set of order [Formula: see text] is a triangle free detour basis of [Formula: see text]. A triangle free detour set [Formula: see text] of [Formula: see text] is called a minimal triangle free detour set if no proper subset of [Formula: see text] is a triangle free detour set of [Formula: see text]. The upper triangle free detour number [Formula: see text] of [Formula: see text] is the maximum order of its minimal triangle free detour sets and any minimal triangle free detour set of order [Formula: see text] is an upper triangle free detour basis of [Formula: see text]. We determine bounds for it and characterize graphs which realize these bounds. For any connected graph [Formula: see text] of order [Formula: see text], [Formula: see text]. Also, for any four positive integers [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] with [Formula: see text], it is shown that there exists a connected graph [Formula: see text] such that [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] is the upper detour number, [Formula: see text] is the upper detour monophonic number and [Formula: see text] is the upper geodetic number of a graph [Formula: see text].

scholarly journals On Riemann surfaces with maximal automorphism groups

1967 ◽  
Vol 8 (2) ◽  
pp. 102-112 ◽  
Author(s):  
Joseph Lehner ◽  
Morris Newman
Keyword(s):  
Automorphism Group ◽  
Riemann Surfaces ◽  
Automorphism Groups ◽  
Universal Covering ◽  
Maximum Order ◽  
Nonabelian Group ◽  
Closed Riemann Surface ◽  
Finite Factor ◽  
Upper Half Plane ◽  
Group 2

Let S be a closed Riemann surface of genusg > 1,so that Ŝ, the universal covering surface of S, is hyperbolic. We can then uniformize S by a discrete, nonabelian group Γ1 of Möbius transformations of the upper half-plane ℋ. It follows that N1 = NΩ(Γ1) is discrete; here N1is the normalizer of Γ in Ω, the group of (conformal) automorphisms of ℋ. An automorphism of S can be lifted to a coset of Nl/Γl. Hence C(S), the group of automorphisms of S, is isomorphic to Nl/Γ1. The order of C = C(S) equals the index of Γ1 in N1, which in turn equals ⃒Γ1⃒ / ⃒Nl⃒, where ⃒Nl⃒ is the hyperbolic area of a fundamental region of Nl. Since Γ1 uniformizes a surface, we have ⃒Γ1⃒ = 4π(g – 1), while, by Siegel's results [7], ⃒N1 ⃒ ≧ π/21 and N1 can only be the triangle group (2, 3, 7). Hence in all cases the order of C(S) is at most 84(g–1), an old result of Hurwitz [1]. The surfaces that permit a maximal automorphism group (= automorphism group of maximum order) can therefore be obtained by studying the finite factor groups of (2, 3, 7). Such a treatment, purely algebraic in nature, has been promised by Macbeath [5].

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.

On quasiprimitive edge-transitive graphs of odd order and twice prime valency

2020 ◽  
Vol 23 (6) ◽  
pp. 1017-1037
Author(s):  
Hong Ci Liao ◽  
Jing Jian Li ◽  
Zai Ping Lu
Keyword(s):  
Automorphism Group ◽  
Connected Graph ◽  
Automorphism Groups ◽  
Primitive Group ◽  
Odd Order ◽  
Vertex Set ◽  
Edge Transitive ◽  
Affine Type ◽  
Transitive Graphs

AbstractA graph is edge-transitive if its automorphism group acts transitively on the edge set. In this paper, we investigate the automorphism groups of edge-transitive graphs of odd order and twice prime valency. Let {\varGamma} be a connected graph of odd order and twice prime valency, and let G be a subgroup of the automorphism group of {\varGamma}. In the case where G acts transitively on the edge set and quasiprimitively on the vertex set of {\varGamma}, we prove that either G is almost simple, or G is a primitive group of affine type. If further G is an almost simple primitive group, then, with two exceptions, the socle of G acts transitively on the edge set of {\varGamma}.

On the Determination of the Maximum Order of the Group of a Tournament

1973 ◽  
Vol 16 (1) ◽  
pp. 11-14 ◽  
Author(s):  
B. Alspach ◽  
J. L. Berggren
Keyword(s):  
Automorphism Group ◽  
Symmetric Group ◽  
Maximum Order ◽  
Odd Order

Let denote the automorphism group of the tournament T. Let g(n) be the maximum of taken over all tournaments of order n. It was noted in [3] that g(n) is also the order of the subgroups of Sn of maximum odd order where Sn denotes the symmetric group of degree n.

scholarly journals THE MOSTAR INDEX OF FULLERENES IN TERMS OF AUTOMORPHISM GROUP

2020 ◽  
Vol 35 (1) ◽  
pp. 151
Author(s):  
Modjtaba Ghorbani ◽  
Shaghayegh Rahmani
Keyword(s):  
Automorphism Group ◽  
Connected Graph ◽  
Topological Index ◽  
Fullerene Graphs

Let $G$ be a connected graph. For an edge $e=uv\in E(G)$, suppose $n(u)$ and $n(v)$ are respectively, the number of vertices of $G$ lying closer to vertex $u$ than to vertex $v$ and the number of vertices of $G$ lying closer to vertex $v$ than to vertex $u$. The Mostar index is a topological index which is defined as $Mo(G)=\sum_{e\in E(G)}f(e)$, where $f(e) = |n(u)-n(v)|$. In this paper, we will compute the Mostar index of a family of fullerene graphs in terms of the automorphism group.  

scholarly journals A Cycle of Maximum Order in a Graph of High Minimum Degree has a Chord

10.37236/7152 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Daniel J. Harvey
Keyword(s):  
General Result ◽  
Connected Graph ◽  
Minimum Degree ◽  
Constant Factor ◽  
Maximum Order ◽  
Chordless Cycle

A well-known conjecture of Thomassen states that every cycle of maximum order in a $3$-connected graph contains a chord. While many partial results towards this conjecture have been obtained, the conjecture itself remains unsolved. In this paper, we prove a stronger result without a connectivity assumption for graphs of high minimum degree, which shows Thomassen's conjecture holds in that case. This result is within a constant factor of best possible. In the process of proving this, we prove a more general result showing that large minimum degree forces a large difference between the order of the largest cycle and the order of the largest chordless cycle.

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