graph

Drawings and pictures are more than mere ornaments in scientific discourse. Blackboard sketches, geological maps, diagrams of molecular structure, astronomical photographs, MRI images, the many varieties of statistical charts and graphs: These pictorial devices are indispensable tools for presenting evidence, for explaining a theory, for telling a story.

Let us take any point of the first graph, for example, x=1/2,y=4/5, that is, the point M_1(1/2,4/5).

Problems involving graphs first appeared in the mathematical folklore as puzzles (e.g. Königsberg bridge problem). Later, graphs appeared in electrical engineering (Kirchhof's Law), chemistry, psychology and economics before becoming a unified field of study.

First, let us define its 1-dimensional analog, that is, a topological graph. A graph 𝛥 is a 1-dimensional stratified topological space with finitely many 0-strata (vertices) and finitely many 1-strata (edges). […] A graph such that any vertex belongs to at least two half-edges we call an s-graph. Clearly the boundary ∂𝛺 of a surface 𝛺 with marked points is an s-graph.

A morphism of graphs 𝜑:𝛥'→𝛥 is a continuous epimorphic map of graphs compatible with the stratification; i.e., the restriction of 𝜑 to any open 1-stratum (interior of an edge) of 𝛥' is a local (therefore, global) homeomorphism with appropriate open 1-stratum of 𝛥.

A graph is a token-level nondistinctive representation of a grapheme. It can differ from the other variants of its grapheme with regard to upper case, lower case, script, print, typeface style, typeface size, etc.