<P> Algebraic graph theory is a branch of mathematics in which algebraic methods are applied to problems about graphs . This is in contrast to geometric, combinatoric, or algorithmic approaches . There are three main branches of algebraic graph theory, involving the use of linear algebra, the use of group theory, and the study of graph invariants . </P> <P> The first branch of algebraic graph theory involves the study of graphs in connection with linear algebra . Especially, it studies the spectrum of the adjacency matrix, or the Laplacian matrix of a graph (this part of algebraic graph theory is also called spectral graph theory). For the Petersen graph, for example, the spectrum of the adjacency matrix is (− 2, − 2, − 2, − 2, 1, 1, 1, 1, 1, 3). Several theorems relate properties of the spectrum to other graph properties . As a simple example, a connected graph with diameter D will have at least D + 1 distinct values in its spectrum . Aspects of graph spectra have been used in analysing the synchronizability of networks . </P> <Table> <Tr> <Th_colspan="5"> Graph families defined by their automorphisms </Th> </Tr> <Tr> <Td> distance - transitive </Td> <Th> → </Th> <Td> distance - regular </Td> <Th> ← </Th> <Td> strongly regular </Td> </Tr> <Tr> <Th> ↓ </Th> <Td> </Td> <Td> </Td> </Tr> <Tr> <Td> symmetric (arc - transitive) </Td> <Th> ← </Th> <Td> t - transitive, t ≥ 2 </Td> <Td> </Td> <Td> skew - symmetric </Td> </Tr> <Tr> <Th> ↓ </Th> <Td> </Td> <Td> </Td> <Td> </Td> <Td> </Td> </Tr> <Tr> <Td> vertex - and edge - transitive </Td> <Th> → </Th> <Td> edge - transitive and regular </Td> <Th> → </Th> <Td> edge - transitive </Td> </Tr> <Tr> <Th> ↓ </Th> <Td> </Td> <Th> ↓ </Th> <Td> </Td> <Th> ↓ </Th> </Tr> <Tr> <Td> vertex - transitive </Td> <Th> → </Th> <Td> regular </Td> <Th> → </Th> <Td> biregular </Td> </Tr> <Tr> <Th> ↑ </Th> <Td> </Td> <Td> </Td> </Tr> <Tr> <Td> Cayley graph </Td> <Th> ← </Th> <Td> zero - symmetric </Td> <Td> </Td> <Td> asymmetric </Td> </Tr> </Table>

Who developed the graph to show algebra geometrically