graphconstruction

Graph construction is the process of creating a graph, a mathematical structure consisting of vertices and edges, to model relationships between objects. It can yield a fixed graph or a graph that evolves over time as objects or relationships change. Construction methods vary with goals, data availability, and constraints.

Common construction methods include data-driven approaches using an edge list or adjacency matrix, and degree-sequence realizations that create a graph with a specified degree sequence (using algorithms such as Havel–Hakimi or Erdős–Gallai). Model-based construction employs predefined rules or distributions, such as random graph models (Erdős–Rényi G(n,p) or G(n,m)) and preferential attachment models (Barabási–Albert), which generate graphs with particular statistical properties.

Other approaches incorporate geometry or topology, including the construction of planar graphs, graphs embedded on surfaces, or graphs derived from incidence structures in combinatorics. Simple, well-known graphs often used in demonstrations include paths, cycles, trees, complete graphs, and bipartite graphs. In practice, a construction task specifies a vertex set and an edge set according to chosen constraints or probabilistic rules.

Implementation typically uses representations such as adjacency lists, adjacency matrices, or incidence structures. After defining the vertex and edge sets, one may verify properties like connectivity, planarity, acyclicity, or specific subgraph patterns. Graph construction also supports sampling, benchmarking, and hypothesis testing in research and application contexts.

Applications of graph construction span computer networks, social networks, chemistry, electrical circuitry, and algorithm testing. It underpins modeling hypotheses, generating benchmark instances, and exploring structural or probabilistic properties of networks. Variants include dynamic or incremental construction, where graphs grow or shrink over time, and graph grammars, which generate graphs through formal production rules.