Topic
Partial geometry
About: Partial geometry is a research topic. Over the lifetime, 50 publications have been published within this topic receiving 586 citations.
Papers
TL;DR: In this survey, it is tried to list all important existence and nonexistence results concerning partial difference sets, e.g., constructions using partial congruence partitions, quadratic forms, cyclotomic classes and finite local rings.
Abstract: LetG be a finite group of order ν. Ak-element subsetD ofG is called a (ν,k, λ, μ)-partial difference set if the expressionsgh
−1, forg andh inD withg≠h, represent each nonidentity element inD exactly λ times and each nonidentity element not inD exactly μ times. Ife∉D andg∈D iffg
−1∈D, thenD is essentially the same as a strongly regular Cayley graph. In this survey, we try to list all important existence and nonexistence results concerning partial difference sets. In particular, various construction methods are studied, e.g., constructions using partial congruence partitions, quadratic forms, cyclotomic classes and finite local rings. Also, the relations with Schur rings, two-weight codes, projective sets, difference sets, divisible difference sets and partial geometries are discussed in detail.
246 citations
TL;DR: In this article, it was shown that there is no strongly regular graph, no two-graph on $96$ vertices, and no partial geometry on $5,9,3 vertices.
Abstract: We show that there is no $(95,40,12,20)$ strongly regular graph and, consequently, there is no $(96,45,24,18)$ strongly regular graph, no two-graph on $96$ vertices, and no partial geometry $\rm{pg}(5,9,3)$. The main idea of the result is based on the star complement technique and requires a small amount of computation.
31 citations
Posted Content•
TL;DR: It is proved that for a fixed integer m>=2, there are only finitely many non-geometric distance-regular graphs with smallest eigenvalue at least -m, diameter at least three and intersection number c"2>= 2.
Abstract: A non-complete geometric distance-regular graph is the point graph of a partial geometry in which the set of lines is a set of Delsarte cliques. In this paper, we prove that for fixed integer $m\geq 2$, there are only finitely many non-geometric distance-regular graphs with smallest eigenvalue at least $-m$, diameter at least three and intersection number $c_2 \geq 2$.
28 citations
TL;DR: An almost complete overview was given on the status of the theory on partial and semipartial geometries in the Handbook of Incidence Geometry in 1995 and this status is updated 5 years later.
27 citations
TL;DR: An efficient method for accelerated analysis of the partial geometry modification problem involving an original structure with small modifications that only requires the solution of the impedance matrix equation of the original structure to be solved.
Abstract: In this paper, we present an efficient method for accelerated analysis of the partial geometry modification problem involving an original structure with small modifications. The conventional method of moments can be very time consuming for this type of problem because the impedance matrix equation has to be solved anew each time the original structure is modified. However, the proposed method only requires the solution of the impedance matrix equation of the original structure. Any small modification of the original object is handled into two steps: first by subtracting a small part from the original structure and then by adding a small part to the rest-structure. Both of these steps can be efficiently computed by using the partitioned-inverse and Sherman–Morrison–Woodbury formulas with the solution of the original structure. When the structure is modified, the proposed method only requires additional operations whose computational burden is $O(N^{2})$ , as opposed to $O(N^{3})$ , where ${N}$ is the number of unknowns in the original structure. Furthermore, the presented method is purely algebraic and rigorous rather than based on approximations. Numerical results for electromagnetic scattering are included in this paper to demonstrate the efficiency and accuracy of this method.
27 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 5 |
| 2020 | 6 |
| 2019 | 1 |
| 2018 | 2 |
| 2016 | 2 |
| 2015 | 1 |