1. What are geometric graphs and their cliques?
Geometric graphs are distance-regular graphs where edges are divided into maximum cliques that reach Delsarte upper bound. These cliques are completely regular codes in distance-regular graphs. Examples of geometric graphs include Hamming, Johnson, Grassmann, dual-polar graphs, etc. Recent papers have reduced the classification problem for completely regular codes in these graphs to simpler graphs. The existence problem for completely regular codes with minimum eigenvalue in any geometric graph is reformulated as that for codes in their clique graph. Completely regular designs in Johnson graphs are interrelated with combinatorial t-designs like Steiner triple and quadruple systems. The classification of completely regular codes in Johnson graphs varies based on strength and covering radius. The incidence relations between codes in Johnson graphs allow for reconstruction of codes in the clique graph. All completely regular codes in Johnson graphs of strength 1 and covering radius 4 are either part of known series or have eigenvalues of 0, 2, or 3. These codes have significant restrictions on their intersection arrays.
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2. What is the relationship between geometric graphs and their clique graphs?
A geometric graph is a graph that is geometric with respect to a set of Delsarte cliques. The clique graph of a geometric graph G with respect to a set of Delsarte cliques K, denoted as G', is a graph where the vertices of G' are the cliques from K, and the edges of G' are the unique cliques that meet in a (w-1)-subset, where w is the diameter of the geometric graph. The clique graph of a geometric graph G with respect to a set of Delsarte cliques K is represented as the Johnson graph J(n, w-1), where n is the number of vertices in the geometric graph and w is the diameter. The incidence of the vertices of the geometric graph and its clique graph is the ordinary subset inclusion relation. The Grassmann graph Jq(n, w) can also be viewed as a geometric graph whose clique graph is Jq(n, w-1).
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3. What defines a completely regular code?
A completely regular code is defined by the adjacency of vertices within the code. Specifically, for a code C, any vertex of C i is adjacent to exactly a i, b i, and g i vertices of C i, C i+1, and C i-1 respectively. The constants a 0, a 1, ..., a r, b 0, b 1, ..., b r-1, and g 1, g 2, ..., g r form the intersection array of the code. This definition allows for the creation of a tridiagonal matrix with specific properties related to the code's structure and eigenvalues.
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4. What are known completely regular codes of strength 1 in Johnson graphs?
Known completely regular codes of strength 1 in Johnson graphs include various constructions based on partitions of sets and disjoint points. For example, GP.1 has p=2 and odd w, w>=5, where C consists of a union of w-1 2 Yi's and one disjoint point. GP.2 has p, q>=3 and w=3, where C consists of a union of i, xYi. GP.3 has q=2 and w=3, where C consists of a union of i, xYi. GP.3' has q=2 and w=4, where C consists of a union of i, xYi. GP.4 has p>=3, q>=2, and w=2, where C consists of a union of i, xYi. GP.5 has p=2, q>=2, and w=2, where C consists of a union of i, xYi. These codes have specific spectra and properties that make them interesting for research in coding theory and graph theory.
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