Hexavalent $(G,s)$-transitive graphs

Song-Tao Guo; Xiao-Hui Hua; Yan-Tao Li

Displaying similar documents to “Hexavalent $(G, s)$ -transitive graphs”

Arc-transitive and s-regular Cayley graphs of valency five on Abelian groups

Mehdi Alaeiyan (2006)

Discussiones Mathematicae Graph Theory

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Let G be a finite group, and let $1_{G} \notin S \subseteq G$ . A Cayley di-graph Γ = Cay(G,S) of G relative to S is a di-graph with a vertex set G such that, for x,y ∈ G, the pair (x,y) is an arc if and only if $y x^{- 1} \in S$ . Further, if $S = S^{- 1} : = s^{- 1} | s \in S$ , then Γ is undirected. Γ is conected if and only if G = ⟨s⟩. A Cayley (di)graph Γ = Cay(G,S) is called normal if the right regular representation of G is a normal subgroup of the automorphism group of Γ. A graph Γ is said to be arc-transitive, if Aut(Γ) is transitive on an arc set. Also,...

Strong Transitivity and Graph Maps

Katsuya Yokoi (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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We study the relation between transitivity and strong transitivity, introduced by W. Parry, for graph self-maps. We establish that if a graph self-map f is transitive and the set of fixed points of $f^{k}$ is finite for each k ≥ 1, then f is strongly transitive. As a corollary, if a piecewise monotone graph self-map is transitive, then it is strongly transitive.

On $k$ -strong distance in strong digraphs

Ping Zhang (2002)

Mathematica Bohemica

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For a nonempty set $S$ of vertices in a strong digraph $D$ , the strong distance $d (S)$ is the minimum size of a strong subdigraph of $D$ containing the vertices of $S$ . If $S$ contains $k$ vertices, then $d (S)$ is referred to as the $k$ -strong distance of $S$ . For an integer $k \geq 2$ and a vertex $v$ of a strong digraph $D$ , the $k$ -strong eccentricity ${s e}_{k} (v)$ of $v$ is the maximum $k$ -strong distance $d (S)$ among all sets $S$ of $k$ vertices in $D$ containing $v$ . The minimum $k$ -strong eccentricity among the vertices of $D$ is its $k$ -strong radius...

Restrained domination in unicyclic graphs

Johannes H. Hattingh, Ernst J. Joubert, Marc Loizeaux, Andrew R. Plummer, Lucas van der Merwe (2009)

Discussiones Mathematicae Graph Theory

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Let G = (V,E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex in V-S is adjacent to a vertex in S and to a vertex in V-S. The restrained domination number of G, denoted by $γ_{r} (G)$ , is the minimum cardinality of a restrained dominating set of G. A unicyclic graph is a connected graph that contains precisely one cycle. We show that if U is a unicyclic graph of order n, then $γ_{r} (U) \geq ⎡ n / 3 ⎤$ , and provide a characterization of graphs achieving this bound.

On Lee's conjecture and some results

Lixia Fan, Zhihe Liang (2009)

Discussiones Mathematicae Graph Theory

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S.M. Lee proposed the conjecture: for any n > 1 and any permutation f in S(n), the permutation graph P(Pₙ,f) is graceful. For any integer n > 1 and permutation f in S(n), we discuss the gracefulness of the permutation graph P(Pₙ,f) if $f = \prod_{k = 0}^{l - 1} (m + 2 k, m + 2 k + 1)$ , and $\prod_{k = 0}^{l - 1} (m + 4 k, m + 4 k + 2) (m + 4 k + 1, m + 4 k + 3)$ for any positive integers m and l.

Distance in stratified graphs

Gary Chartrand, Lisa Hansen, Reza Rashidi, Naveed Sherwani (2000)

Czechoslovak Mathematical Journal

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A graph $G$ is stratified if its vertex set is partitioned into classes, called strata. If there are $k$ strata, then $G$ is $k$ -stratified. These graphs were introduced to study problems in VLSI design. The strata in a stratified graph are also referred to as color classes. For a color $X$ in a stratified graph $G$ , the $X$ -eccentricity $e_{X} (v)$ of a vertex $v$ of $G$ is the distance between $v$ and an $X$ -colored vertex furthest from $v$ . The minimum $X$ -eccentricity among the vertices of $G$ is the $X$ -radius ${r a d}_{X} G$ of $G$ ...

More classes of non-orbit-transitive operators

Carl Pearcy, Lidia Smith (2010)

Studia Mathematica

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In [JKP] and its sequel [FPS] the authors initiated a program whose (announced) goal is to eventually show that no operator in ℒ(ℋ) is orbit-transitive. In [JKP] it is shown, for example, that if T ∈ ℒ(ℋ) and the essential (Calkin) norm of T is equal to its essential spectral radius, then no compact perturbation of T is orbit-transitive, and in [FPS] this result is extended to say that no element of this same class of operators is weakly orbit-transitive. In the present note we show...

Signed total domination number of a graph

Bohdan Zelinka (2001)

Czechoslovak Mathematical Journal

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The signed total domination number of a graph is a certain variant of the domination number. If $v$ is a vertex of a graph $G$ , then $N (v)$ is its oper neighbourhood, i.e. the set of all vertices adjacent to $v$ in $G$ . A mapping $f : V (G) \to {- 1, 1}$ , where $V (G)$ is the vertex set of $G$ , is called a signed total dominating function (STDF) on $G$ , if $\sum_{x \in N (v)} f (x) \geq 1$ for each $v \in V (G)$ . The minimum of values $\sum_{x \in V (G)} f (x)$ , taken over all STDF’s of $G$ , is called the signed total domination number of $G$ and denoted by $γ_{s t} (G)$ . A theorem stating lower bounds for $γ_{s t} (G)$ is...

Graphs with large double domination numbers

Michael A. Henning (2005)

Discussiones Mathematicae Graph Theory

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In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V(G) is a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality of a double dominating set of G is the double domination number $γ_{\times 2} (G)$ . If G ≠ C₅ is a connected graph of order n with minimum degree at least 2, then we show that $γ_{\times 2} (G) \leq 3 n / 4$ and we characterize those graphs achieving equality.

Displaying similar documents to “Hexavalent ( G , s ) -transitive graphs”

Displaying similar documents to “Hexavalent $(G, s)$ -transitive graphs”