Generating Graphs with Symmetry

Symmetry in Generating Functions

Generating functions play important roles in theory of orthogonal polynomials. In particular, it is important to consider generating functions that have symmetry. This paper is a survey on generating functions that define unitary operators. First, classical generating functions that define unitary operators are discussed. Next, group theoretical approach to generating functions that have unitar...

Generating Graphs with Guarantees on Partition Costs

Partitioning a graph is to divide its set of vertices into a fixed number of equally sized disjoint subsets minimizing the number of edges between them. In this work, we describe an algorithm to generate large random graphs such that an optimal solution to this problem is known. For this task, we first generate dense subgraphs such that we know a lower bound on the size of the minimal cut for e...

Generating Graphs with Predefined k-Core Structure∗

The modeling of realistic networks is of great importance for modern complex systems research. Previous procedures typically model the natural growth of networks by means of iteratively adding nodes, geometric positioning information, a definition of link connectivity based on the preference for nearest neighbors or already highly connected nodes, or combine several of these approaches. Our nov...

Generating random graphs with large girth

We present a simple and efficient algorithm for randomly generating simple graphs without small cycles. These graphs can be used to design high performance Low-Density Parity-Check (LDPC) codes. For any constant k, α ≤ 1/2k(k + 3) and m = O(n), our algorithm generates an asymptotically uniform random graph with n vertices, m edges, and girth larger than k in polynomial time. To the best of our ...

Generating Random Graphs with Tunable Clustering Coefficient