Algebraic graph theory
Algebraic graph theory studies graphs using algebra and, less commonly, algebra using graph theory. Major subtopics include graph homomorphisms and automorphisms, spectral graph theory, and polynomial graph invariants.
General references on algebraic graph theory are:
- Biggs, 1993: Algebraic graph theory, 2nd ed. (doi)
- Now somewhat dated, but the original 1974 edition was the first significant book on algebraic graph theory
- Listed by Richard Lipton has an all-time favorite math book
- Godsil & Royle, 2001: Algebraic graph theory (doi)
Morphisms of graphs
For the believing structuralist, graphs should be studied through their morphisms, which are graph homomorphisms.
- Hahn & Tardif, 1997: Graph homomorphisms: structure and symmetry (doi, pdf)
- Hahn & MacGillivray, 2002, manuscript: Graph homomorphisms: Computational aspects and infinite graphs
- Godsil & Royle, 2001: Algebraic graph theory, Ch. 6: Homomorphisms (doi)
- Hell & Nesetril, 2004: Graphs and homomorphisms (doi)
The automorphism group of a graph captures its symmetry, or lack thereof.
- Biggs, 1993: Algebraic graph theory, 2nd ed., Part 3: Symmetry and regularity
- Godsil & Royle, 2001: Algebraic graph theory, Ch. 2: Groups (doi)
Ronnie Brown, following his former student John Shrimpton, describes an automorphism graph, whose vertices make up the automorphism group.
Spectral graph theory
Standard references on spectral graph theory are:
- Cvetković, Doob, Sachs, 1980: Spectra of graphs
- Cvetković, Rowlinson, Simić, 1997: Eigenspaces of graphs
- Chung, 1997: Spectral graph theory (doi, online )
More recent books include: