Origins
The English word 'graph' in its modern mathematical sense is a product of the nineteenth century, but its Greek ancestry reaches back to some of the most fundamental acts of civilization: writing, drawing, and recording.
Greek 'graphein' (γράφειν) meant 'to scratch, to carve, to write, to draw.' The original physical sense was scratching marks into a surface — wax tablets, clay, stone — and from this concrete act grew the abstract concept of writing itself. The noun 'graphē' (γραφή) meant both 'writing' and 'drawing,' a dual meaning that reflects the ancient world's lack of a sharp distinction between the two activities. In Athens, 'graphē' also meant a written legal accusation — the act of inscribing a formal charge.
The PIE root *gerbh- (to scratch, carve) connects 'graphein' to a small family of words across Indo-European. In English, the Germanic descendant is 'carve,' from Old English 'ceorfan' (to cut, carve), though the connection is debated by some etymologists. What is certain is that the Greek branch of this root became one of the most productive sources of English vocabulary.
Word Formation
The suffix '-graph' (an instrument that writes or records) and '-graphy' (the process or art of writing/recording) appear in an extraordinary number of English words. 'Photograph' is writing with light. 'Telegraph' is writing from afar. 'Biography' is life-writing. 'Geography' is earth-writing. 'Calligraphy' is beautiful writing. 'Autograph' is self-writing. 'Paragraph' is writing beside (a marginal mark). 'Seismograph' records earthquakes. 'Phonograph' records sound. The pattern is enormously productive and continues to generate new terms.
The mathematical sense of 'graph' — a diagram showing the relationship between quantities — was established in the 1870s. The term appears to have been popularized by the mathematician James Joseph Sylvester, who in 1878 used 'graph' in the context of chemical diagrams and mathematical relations. The word was either a back-formation from 'graphic' (pertaining to drawing, from Latin 'graphicus,' from Greek 'graphikós') or a direct shortening of 'graphic formula.'
Graph theory, the mathematical study of networks of nodes and edges, also uses 'graph' in a related but distinct sense. Here a graph is not a plotted curve but an abstract structure of vertices and connections. This usage dates to Leonhard Euler's 1736 paper on the Seven Bridges of Konigsberg, though the term 'graph' for such structures was formalized later.
Latin Roots
The adjective 'graphic' entered English in the seventeenth century from Latin 'graphicus' (of or pertaining to drawing or painting), which came directly from Greek 'graphikós.' The sense 'vivid, producing a strong clear impression' developed from the idea that good writing or description creates a picture in the mind — a meaning that connects the word's dual heritage of writing and drawing.
In modern English, the '-graph' and '-graphy' family numbers well over a hundred established terms, making Greek 'graphein' one of the single most productive roots in the English language — rivaled only by Latin 'scribere' (to write) in the domain of writing-related vocabulary.