Edge growth in graph powers

For a graph G, its rth power G^r has the same vertex set as G, and has an edge between any two vertices within distance r of each other in G. We give a lower bound for the number of edges in the rth power of G in terms of the order of G and the minimal degree of G. As a corollary we determine how small the ratio e(G^r)/e(G) can be for regular graphs of diameter at least r

Edge growth in graph powers

For a graph G, its rth power G^r has the same vertex set as G, and has an edge between any two vertices within distance r of each other in G. We give a lower bound for the number of edges in the rth power of G in terms of the order of G and the minimal degree of G. As a corollary we determine how small the ratio e(G^r)/e(G) can be for regular graphs of diameter at least r

Rate-distance tradeoff for codes above graph capacity

The capacity of a graph is defined as the rate of exponential growth of independent sets in the strong powers of the graph. In the strong power an edge connects two sequences if at each position their letters are equal or adjacent. We consider a variation of the problem where edges in the power graphs are removed between sequences which differ in more than a fraction $\delta$ of coordinates. The proposed generalization can be interpreted as the problem of determining the highest rate of zero undetected-error communication over a link with adversarial noise, where only a fraction $\delta$ of symbols can be perturbed and only some substitutions are allowed. We derive lower bounds on achievable rates by combining graph homomorphisms with a graph-theoretic generalization of the Gilbert-Varshamov bound. We then give an upper bound, based on Delsarte's linear programming approach, which combines Lov\'asz' theta function with the construction used by McEliece et al. for bounding the minimum distance of codes in Hamming spaces.Comment: 5 pages. Presented at 2016 IEEE International Symposium on Information Theor

Desain Dan Analisis Algoritma Penyelesaian Persoalan SPOJ Maximum Edge Of Powers Of Permutation Dengan Metode Permutation Cycles Finding Dan FFT Convolution

Permasalahan dalam buku tugas akhir ini adalah permasalahan “Maximum Edge of Powers of Permutation” yang terdapat pada situs penilaian daring Sphere Online Judge(SPOJ) dengan nomor soal 6895 dan kode soal MEPPERM. Dalam permasalahan ini, diberikan sejumlah simpul yang masing-masing memiliki bobot keluar dan masuk. Diberikan suatu permutasi terhadap sejumlah simpul tersebut. Suatu graf dibentuk dari setiap simpul yang terhubung ke simpul hasil permutasi dengan bobot jumlah bobot keluar simpul asal dan bobot masuk simpul tujuan. Dicari sisi dengan bobot terbesar pada setiap graf yang dibentuk dari hasil pangkat permutasi. Tugas akhir ini akan mengimplementasikan metode pencarian permutasi siklus dan konvolusi menggunakan transformasi Fourier cepat(FFT) dalam menyelesaikan permasalahan Maximum Edge of Powers of Permutation. Metode transformasi Fourier cepat yang digunakan adalah algoritma Cooley-Tukey. Implementasi dalam tugas akhir ini menggunakan bahasa pemrograman C++. Hasil uji coba menunjukkan bahwa sistem menghasilkan bobot sisi maksimum pada setiap pangkat permutasi dengan benar dan memiliki pertumbuhan waktu dengan kompleksitas O(NMlogNM+Q*sqrt(N)). ===================================================================================== This final project is based on “Maximum Edge of Powers of Permutation” problem on SPOJ with problem number 6895 and problem code MEPPERM. In this problem, given vertices which each have weight in and out. Given a permutation for those vertices. A graph can be built by connecting each vertices to its permutation with edge weight sum of vertex weight out and vertex weight in from its permutation. Determine edge with maximum weight from each graph which are created from the powers of permutation. This final project implements permutation cycles finding and convolution using fast Fourier Transform(FFT) to solve the problem of Maximum Edge of Powers of Permutation. Fast Fourier transform method used is Cooley-Tukey algorithm. The implementation of final project uses C++ programming language. The experiment result proved the system provide edge with maximum weight for each powers of permutation correctly and has growth time with complexity of O(NMlogNM+Q*sqrt(N))

Growth of graph powers

For a graph G, its rth power is constructed by placing an edge between two vertices if they are within distance r of each other. In this note we study the amount of edges added to a graph by taking its rth power. In particular we obtain that either the rth power is complete or "many" new edges are added. This is an extension of a result obtained by P. Hegarty for cubes of graphs.Comment: 6 pages, 1 figur

Growth and order of automorphisms of free groups and free Burnside groups

We prove that an outer automorphism of the free group is exponentially growing if and only if it induces an outer automorphism of infinite order of free Burnside groups with sufficiently large odd exponent.Comment: 36 pages, 4 figure

Negative curvature in graphical small cancellation groups

We use the interplay between combinatorial and coarse geometric versions of negative curvature to investigate the geometry of infinitely presented graphical $Gr'(1/6)$ small cancellation groups. In particular, we characterize their 'contracting geodesics', which should be thought of as the geodesics that behave hyperbolically. We show that every degree of contraction can be achieved by a geodesic in a finitely generated group. We construct the first example of a finitely generated group $G$ containing an element $g$ that is strongly contracting with respect to one finite generating set of $G$ and not strongly contracting with respect to another. In the case of classical $C'(1/6)$ small cancellation groups we give complete characterizations of geodesics that are Morse and that are strongly contracting. We show that many graphical $Gr'(1/6)$ small cancellation groups contain strongly contracting elements and, in particular, are growth tight. We construct uncountably many quasi-isometry classes of finitely generated, torsion-free groups in which every maximal cyclic subgroup is hyperbolically embedded. These are the first examples of this kind that are not subgroups of hyperbolic groups. In the course of our analysis we show that if the defining graph of a graphical $Gr'(1/6)$ small cancellation group has finite components, then the elements of the group have translation lengths that are rational and bounded away from zero.Comment: 40 pages, 14 figures, v2: improved introduction, updated statement of Theorem 4.4, v3: new title (previously: "Contracting geodesics in infinitely presented graphical small cancellation groups"), minor changes, to appear in Groups, Geometry, and Dynamic