University of Illinois Urbana-Champaign

Polynomials of the Adjacency Matrix of a Graph (distance-Transitive, Distance-Regular, Orbit)

Beezer, Robert Arnold

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Description

Title
Polynomials of the Adjacency Matrix of a Graph (distance-Transitive, Distance-Regular, Orbit)
Author(s)
Beezer, Robert Arnold
Issue Date
1984
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Abstract
  • Given graphs (GAMMA) and (DELTA), and a real polynomial r(x), we will say that (DELTA) is generated from (GAMMA) by r(x) if r(A((GAMMA))) = A((DELTA)) where A((GAMMA)) and A((DELTA)) are adjacency matrices. For several interesting classes of graphs it is possible to determine all of the graphs which can be generated by a polynomial. Define the ith distance graph, (GAMMA)(,i), as the graph with the same vertex set as (GAMMA) and two vertices are adjacent in (GAMMA)(,i) if and only if they are a distance i apart. If (GAMMA) is a distance-regular graph, then for each i there exists a polynomial of degree i, p(,i)(x), such that p(,i)(A((GAMMA))) = A((GAMMA)(,i)). In fact, it has been shown that this property characterizes distance-regular graphs.
  • With the above situation in mind, we construct the definition of an orbit polynomial graph. A graph is orbit polynomial if certain natural 0-1 matrices (determined by the automorphism group of the graph) are equal to polynomials of the adjacency matrix of the graph. We obtain many results about the properties of these graphs and their connections with association schemes. We also characterize orbit polynomial graphs with a prime number of vertices and the non-symmetric trivalent orbit polynomial graphs.
  • We then study the graphs generated from a tree by a polynomial. For a path, all of the possible graphs are determined. A sunset is a path of even length with additional vertices adjacent to the central vertex. We produce a polynomial q(x) which generates a graph from a sunset which happens to be isomorphic to the original sunset. Motivated by this example, we study the situation where r(A((GAMMA))) = A((DELTA)), r(x) (NOT=) x, and (GAMMA) is isomorphic to (DELTA).
Graduation Semester
1984
Type of Resource
text
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http://hdl.handle.net/2142/71218

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