H-E-Super Magic Decomposition of Graphs

An H-magic labeling in an H-decomposable graph G is a bijection f:V(G) U E(G) --> {1,2, … ,p+q} such that for every copy H in the decomposition, $\sum\limits_{v\in V(H)} f(v)+\sum\limits_{e\in E(H)} f(e)$ is constant. The function f is said to be H-E-super magic if f(E(G)) = {1,2, … ,q}. In this paper, we study some basic properties of m-factor-E-super magic labelingand we provide a necessary and sufficient condition for an even regular graph to be 2-factor-E-super magic decomposable. For this purpose, we use Petersen\u27s theorem and magic squares

PELABELAN TOTAL CYCLE PELABELAN TOTAL SUPER VERTEX-MAGIC PADA CYCLE DAN GRAF CIRCULANT PADA

Nony Oktavy Liliyani, 2010, PELABELAN TOTAL SUPER VERTEX- MAGIC PADA CYCLE DAN GRAF CIRCULANT. Fakultas Matematika dan Ilmu Pengetahuan Alam, Universitas Sebelas Maret Surakarta. Pelabelan graf adalah pemberian label pada vertex, edge atau vertex sekaligus edge. Pemberian label pada vertex sekaligus edge disebut pelabelan total. 껐 ൌ ሺ ܸ ,ܧ ሻ menyatakan sebuah graf berhingga, sederhana dan tak berarah, dengan V dan E masing-masing adalah himpunan vertex dan edge dalam 껐. Diasumsikan N(ݒ ௜ ) adalah himpunan vertex di persekitaran ݒ ௜א껐, ) adalah order dan ε adalah size dalam graf 껐. Pelabelan total vertex-magic adalah suatu bijeksi λ : ܸ׫ܧ՜ ሼ 1,2,…, )൅ߝ ሽ dengan syarat bahwa untuk setiap ݒ ௜אܸ ሺ 껐ሻ berlaku ߣ ሺ ݒ ௜ ሻ ൅ ෍ ߣ ሺ ݒ ௜ ݒ ̊ሻ ௩ ೕ 銐 ሺ ௩ ೔ ሻ ൌ݇ dengan k adalah konstanta magic yang bernilai konstan. Pelabelan total vertex- magic disebut super jika λ(V) = {1, 2, … , υ}. Graf yang memuat pelabelan total super vertex-magic disebut graf super vertex-magic. Graf yang digunakan sebagai objek penulisan skripsi adalah cycle ú௡ , gabungan disjoint m cycle ذú௡ , graf circulant ú௡ ሺ 1,ݏ ሻ , ú௡ ሺ 1,2,3ሻ , ú௡ ሺ 1,2,3,4ሻ dan gabungan disjoint m graf circulant ذú௡ ሺ 1,ݏ ሻ . Pembahasan skripsi merupakan kaji ulang jurnal yang bertujuan mengetahui cycle dan graf circulant yang memuat pelabelan total super vertex magic, mengetahui pelabelan total super vertex-magic pada graf-graf objek penulisan. Metode penulisan yang digunakan adalah studi literatur. Kesimpulan dari hasil pembahasan skripsi adalah sebagai berikut. 1. Pelabelan total super vertex-magic termuat dalam cycle ú௡ dan graf circulant ú௡ ሺ 1,2,ڮ,ሺ tെ1ሻ2 ⁄ ሻ yang memiliki n ganjil. Gabungan disjoint m cycle ذú௡ dan gabungan disjoint m graf circulant ذú௡ ሺ 1,2,ڮ,ሺ tെ1ሻ2 ⁄ ሻ mempunyai pelabelan total super vertex-magic jika ذ dan t ganjil. 2. Konstanta magic pada pelabelan total super vertex-magic ditentukan dengan rumus ݇ൌ ሺ )൅ߝ ሻ ሺ )൅ߝ ൅1ሻ ) െሺ )൅1ሻ 2 Ǥ Pelabelan graf dilakukan dengan aturan tertentu sedemikian hingga dihasilkan konstanta magic k. Kata kunci : pelabelan magic, pelabelan total super vertex-magic, cycle, graf circulant

Connectivity and other invariants of generalized products of graphs

Figueroa-Centeno et al. [4] introduced the following product of digraphs let D be a digraph and let G be a family of digraphs such that V (F) = V for every F¿G. Consider any function h:E(D)¿G. Then the product D¿hG is the digraph with vertex set V(D)×V and ((a,x),(b,y))¿E(D¿hG) if and only if (a,b)¿E(D) and (x,y)¿E(h(a,b)). In this paper, we deal with the undirected version of the ¿h-product, which is a generalization of the classical direct product of graphs and, motivated by the ¿h-product, we also recover a generalization of the classical lexicographic product of graphs, namely the °h-product, that was introduced by Sabidussi in 1961. We provide two characterizations for the connectivity of G¿hG that generalize the existing one for the direct product. For G°hG, we provide exact formulas for the connectivity and the edge-connectivity, under the assumption that V (F) = V , for all F¿G. We also introduce some miscellaneous results about other invariants in terms of the factors of both, the ¿h-product and the °h-product. Some of them are easily obtained from the corresponding product of two graphs, but many others generalize the existing ones for the direct and the lexicographic product, respectively. We end up the paper by presenting some structural properties. An interesting result in this direction is a characterization for the existence of a nontrivial decomposition of a given graph G in terms of ¿h-product.Postprint (author's final draft

VoG: Summarizing and Understanding Large Graphs

How can we succinctly describe a million-node graph with a few simple sentences? How can we measure the "importance" of a set of discovered subgraphs in a large graph? These are exactly the problems we focus on. Our main ideas are to construct a "vocabulary" of subgraph-types that often occur in real graphs (e.g., stars, cliques, chains), and from a set of subgraphs, find the most succinct description of a graph in terms of this vocabulary. We measure success in a well-founded way by means of the Minimum Description Length (MDL) principle: a subgraph is included in the summary if it decreases the total description length of the graph. Our contributions are three-fold: (a) formulation: we provide a principled encoding scheme to choose vocabulary subgraphs; (b) algorithm: we develop \method, an efficient method to minimize the description cost, and (c) applicability: we report experimental results on multi-million-edge real graphs, including Flickr and the Notre Dame web graph.Comment: SIAM International Conference on Data Mining (SDM) 201