#### Date of Award:

2016

#### Document Type:

Dissertation

#### Degree Name:

Doctor of Philosophy (PhD)

#### Department:

Mathematics and Statistics

#### Advisor/Chair:

David Brown

#### Abstract

We will introduce three new classes of graphs; namely bipartite dot product graphs, probe dot product graphs, and combinatorial orthogonal graphs. All of these representations were inspired by a vector representation known as a dot product representation.

Given a bipartite graph G = (X, Y, E), the bipartite dot product representation of G is a function ƒ : X ∪ Y → R^{k} and a positive threshold *t* such that for any κ ∈ Χ and γ ∈ Υ , κγ ∈ ε if and only if f(κ) · f(γ) ≥ *t*. The minimum *k* such that a bipartite dot product representation exists for G is the bipartite dot product dimension of G, denoted *bdp*(G). We will show that such representations exist for all bipartite graphs as well as give an upper bound for the bipartite dot product dimension of any graph. We will also characterize the bipartite graphs of bipartite dot product dimension 1 by their forbidden subgraphs.

An undirected graph G = (V, E) is a probe C graph if its vertex set can be parti-tioned into two sets, N (nonprobes) and P (probes) where N is independent and there exists E' ⊆ N × N such that G' = (V, E ∪ E) is a C graph. In this dissertation we introduce probe *k*-dot product graphs and characterize (at least partially) probe 1-dot product graphs in terms of forbidden subgraphs and certain 2-SAT formulas. These characterizations are given for the very different circumstances: when the partition into probes and nonprobes is given, and when the partition is not given.

Vectors κ = (κ_{1}, κ_{2}, . . . , κ_{n})^{T} and γ = (γ_{1}, γ_{2}, . . . , γ_{n})^{T} are combinatorially orthogonal if |{*i* : κ_{i}γ_{i} = 0}| ≠ 1. An undirected graph G = (V, E) is a combinatorial orthogonal graph if there exists ƒ : V → R^{n} for some *n* ∈ Ν such that for any *u*, υ &Isin; V , *uv* ∉ E iff ƒ(*u*) and ƒ(*v*) are combinatorially orthogonal. These representations can also be limited to a mapping *g* : V → {0, 1}^{n} such that for any *u,v* ∈ V , *uv* ∉ E iff *g(u) · g(v)* = 1. We will show that every graph has a combinatorial orthogonal representation. We will also state the minimum dimension necessary to generate such a representation for specific classes of graphs.

#### Checksum

6731fb5d19d55de675a461e4fe6b8d98

#### Recommended Citation

Bailey, Sean, "To Dot Product Graphs and Beyond" (2016). *All Graduate Theses and Dissertations*. 5029.

https://digitalcommons.usu.edu/etd/5029

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