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Spectral graph theory emerged in the 1950s and 1960s. Besides graph theoretic research on the relationship between structural and spectral properties of graphs, another major source was research in quantum chemistry , but the connections between these two lines of work were not discovered until much later. [ 15 ]
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.
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. [1]
The Laplacian matrix is the easiest to define for a simple graph, but more common in applications for an edge-weighted graph, i.e., with weights on its edges — the entries of the graph adjacency matrix. Spectral graph theory relates properties of a graph to a spectrum, i.e., eigenvalues, and eigenvectors of matrices associated with the graph ...
As Murty's survey paper [1] notes, Ramanujan graphs "fuse diverse branches of pure mathematics, namely, number theory, representation theory, and algebraic geometry". These graphs are indirectly named after Srinivasa Ramanujan; their name comes from the Ramanujan–Petersson conjecture, which was used in a construction of some of these graphs.
In spectral graph theory, the Alon–Boppana bound provides a lower bound on the second-largest eigenvalue of the adjacency matrix of a -regular graph, [1] meaning a graph in which every vertex has degree .
Algebraic graph theory is a branch of graph theory ... Spectral graph theory; A. Adjacency algebra; Adjacency matrix; Algebraic connectivity; Alon–Boppana bound;
The relationship between a graph and the eigenvalues and eigenvectors of its adjacency matrix is studied in spectral graph theory. The adjacency matrix of a graph should be distinguished from its incidence matrix, a different matrix representation whose elements indicate whether vertex–edge pairs are incident or not, and its degree matrix ...