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The Brouwer–Haemers graph is the first in an infinite family of Ramanujan graphs defined as generalized Paley graphs over fields of characteristic three. [2] With the 3 × 3 {\displaystyle 3\times 3} Rook's graph and the Games graph , it is one of only three possible strongly regular graphs whose parameters have the form ( ( n 2 + 3 n − 1 ...
Brouwer has confirmed by computation that the conjecture is valid for all graphs with at most 10 vertices. [1] It is also known that the conjecture is valid for any number of vertices if t = 1, 2, n − 1, and n. For certain types of graphs, Brouwer's conjecture is known to be valid for all t and for any number of vertices
Andries Brouwer and Hendrik van Maldeghem (see #References) use an alternate but fully equivalent definition of a strongly regular graph based on spectral graph theory: a strongly regular graph is a finite regular graph that has exactly three eigenvalues, only one of which is equal to the degree k, of multiplicity 1.
The 1980 monograph Spectra of Graphs [16] by Cvetković, Doob, and Sachs summarised nearly all research to date in the area. In 1988 it was updated by the survey Recent Results in the Theory of Graph Spectra. [17] The 3rd edition of Spectra of Graphs (1995) contains a summary of the further recent contributions to the subject. [15]
A complete bipartite graph of K 4,7 showing that Turán's brick factory problem with 4 storage sites (yellow spots) and 7 kilns (blue spots) requires 18 crossings (red dots) For any k, K 1,k is called a star. [2] All complete bipartite graphs which are trees are stars. The graph K 1,3 is called a claw, and is used to define the claw-free graphs ...
In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree of each internal vertex are equal to each other. [ 1 ]
In mathematics, the energy of a graph is the sum of the absolute values of the eigenvalues of the adjacency matrix of the graph. This quantity is studied in the context of spectral graph theory. More precisely, let G be a graph with n vertices. It is assumed that G is a simple graph, that is, it does not
The layout uses the eigenvectors of a matrix, such as the Laplace matrix of the graph, as Cartesian coordinates of the graph's vertices. The idea of the layout is to compute the two largest (or smallest) eigenvalues and corresponding eigenvectors of the Laplacian matrix of the graph and then use those for actually placing the nodes.