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Journal of the London Mathematical Society, 1986
This paper completes the classification of cubic distance-regular graphs. We define the profile and period of certain cycles in such a graph, and obtain congruence conditions on the periods that help determine the feasible intersection arrays. It turns out that there are just 13 possible cases and in each case the array is realised by a unique graph.
Biggs, N.L. +2 more
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This paper completes the classification of cubic distance-regular graphs. We define the profile and period of certain cycles in such a graph, and obtain congruence conditions on the periods that help determine the feasible intersection arrays. It turns out that there are just 13 possible cases and in each case the array is realised by a unique graph.
Biggs, N.L. +2 more
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On Bounding the Diameter of a Distance-Regular Graph
Combinatorica, 2021It is investigated how to use an initial portion of the intersection array of a distance-regular graph to give an upper bound for the diameter of the graph. In particular, three new diameter bounds are obtained. These bounds are tight for the Hamming \(d\)-cube, doubled odd graphs, the Heawood graph, Tutte's \(8\)-cage and \(12\)-cage, the generalized ...
Arnold Neumaier, Safet Penjic
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Eigenpolytopes of Distance Regular Graphs
Canadian Journal of Mathematics, 1998AbstractLet X be a graph with vertex set V and let A be its adjacency matrix. If E is the matrix representing orthogonal projection onto an eigenspace of A with dimension m, then E is positive semi-definite. Hence it is the Gram matrix of a set of |V| vectors in Rm. We call the convex hull of a such a set of vectors an eigenpolytope of X.
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Eigenvectors of Distance-Regular Graphs
SIAM Journal on Matrix Analysis and Applications, 1988The author studies the set of points the coordinates of which are rows of the matrix in which the columns are orthogonal eigenvectors associated to an eigenvalue of the adjacency matrix of a graph. In particular, the second largest eigenvalue \(\alpha\) and distance-regular graphs G are considered.
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Distance-regular Graphs of the Height h
Graphs and Combinatorics, 1999zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Bounding the diameter of distance-regular graphs
Combinatorica, 1988Let G be a connected distance regular graph with valence \(k>2\) and diameter d. Suppose further that G is not a complete multipartite graph. let \(\theta\) be an eigenvalue of G with \(\theta \neq Ik,\) and \(m>1\). Then there are only finitely many connected, co-connected distance regular graphs with an eigenvalue of multiplicity m.
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A Bound for the Diameter of Distance-Regular Graphs
Combinatorica, 1999zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Potential Theory on Distance-Regular Graphs
Combinatorics, Probability and Computing, 1993A graph may be regarded as an electrical network in which each edge has unit resistance. We obtain explicit formulae for the effective resistance of the network when a current enters at one vertex and leaves at another in the distance-regular case.
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1978
Inequalities are obtained between the various parameters of a distance-regular graph. In particular, if k1 is the valency and k2 is the number of vertices at distance two from a given vertex, then in general k1 ⩽ k2. For distance-regular graphs of diameter at least four, k1=k2 if and only if the graph is simply a circuit.
D. E. Taylor, Richard Levingston
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Inequalities are obtained between the various parameters of a distance-regular graph. In particular, if k1 is the valency and k2 is the number of vertices at distance two from a given vertex, then in general k1 ⩽ k2. For distance-regular graphs of diameter at least four, k1=k2 if and only if the graph is simply a circuit.
D. E. Taylor, Richard Levingston
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