Grup Automorfisma Graf Tangga dan Graf Lingkaran

Abstrak. Automorfisma dari suatu graf G merupakan isomorfisma dari graf G ke dirinya sendiri, yaitu fungsi yang memetakan dirinya sendiri. Automorfisma suatu graf dapat dicari dengan menentukan semua kemungkinan fungsi yang satu-satu, onto serta isomorfisma dari himpunan titik pada graf tersebut. Artikel ini difokuskan pada penentuan banyaknya fungsi pada graf tangga dan graf lingkaran yang automorfisma serta grup yang dibentuk oleh himpunan automorfisma dari kedua graf tersebut. Jenis penelitian ini merupakan penelitian dasar atau penelitian murni dan metode yang digunakan adalah studi literatur. Hasil penelitian ini menunjukkan bahwa graf tangga membentuk grup abelian berorde-2, graf tangga membentuk grup dihedral berorde-8, dan graf tangga membentuk grup abelian berorde-4. Sedangkan graf lingkaran membentuk grup dihedral berorde-2n.Kata Kunci: Automorfisma, Graf Lingkaran, Graf Tangga, GrupAbstract. An automorphism of a graph G is an isomorphism of graph G to itself i.e. the function that maps onto itself. An automorphism of a graph can be looked for by determining all possible functions which is one-to-one, onto, and isomorphism from vertex set at the graph. This article is focused on determining the number of automorphism functions on ladder graph and cycle graph and the groups formed by the two graphs. The tipe of this research is basic research or pure research and the research method used is literarture review. The result show that ladder graph forms an abelian group of order 2, ladder graph forms a dihedral group of order 8, and ladder graph forms an abelian group of order 4. In other side, cycle graph , forms a dihedral group of order 2n.Keywords: Automorphism, Cycle Graph, Ladder Graph, Group

Removing Symmetry in Circulant Graphs and Point-Block Incidence Graphs

An automorphism of a graph is a mapping of the vertices onto themselves such that connections between respective edges are preserved. A vertex v in a graph G is fixed if it is mapped to itself under every automorphism of G. The fixing number of a graph G is the minimum number of vertices, when fixed, fixes all of the vertices in G. The determination of fixing numbers is important as it can be useful in determining the group of automorphisms of a graph-a famous and difficult problem. Fixing numbers were introduced and initially studied by Gibbons and Laison, Erwin and Harary and Boutin. In this paper, we investigate fixing numbers for graphs with an underlying cyclic structure, which provides an inherent presence of symmetry. We first determine fixing numbers for circulant graphs, showing in many cases the fixing number is 2. However, we also show that circulant graphs with twins, which are pairs of vertices with the same neighbourhoods, have considerably higher fixing numbers. This is the first paper that investigates fixing numbers of point-block incidence graphs, which lie at the intersection of graph theory and combinatorial design theory. We also present a surprising result-identifying infinite families of graphs in which fixing any vertex fixes every vertex, thus removing all symmetries from the graph.

CONJUGATING AUTOMORPHISMS OF GRAPH PRODUCTS: KAZHDAN’S PROPERTY (T) AND SQ-UNIVERSALITY

An automorphism of a graph product of groups is conjugating if it sends each factor to a conjugate of a factor (possibly different). In this article, we determine precisely when the group of conjugating automorphisms of a graph product satisfies Kazhdan’s property (T) and when it satisfies some vastness properties including SQ-universality.

EDGE-TRANSITIVE GRAPH AUTOMORPHISMS AND PERIODIC SURFACE HOMEOMORPHISMS

An automorphism of a graph is edge-transitive if it acts transitively on the set of geometric edges (components of the complement of the vertices), or equivalently, if there is no nontrivial invariant subgraph. Every such automorphism can be embedded as the restriction to an invariant spine of some orientation-preserving periodic homeomorphism of a punctured surface. We find all the edge-transitive graph automorphisms and for each, find a complete list (up to a natural equivalence relation) of the possible ways that it can be embedded in a periodic homeomorphism.