#### Event Title

Finding a Uniformly Most Reliable Graph

#### Faculty Mentor

Dr. Christina Graves

#### Document Type

Oral Presentation

#### Date of Publication

April 2021

#### Abstract

The reliability polynomial of a simple graph G represents the probability that G will remain connected given a fixed probability 1 − p of each edge failing. A graph G is uniformly most reliable if its reliability polynomial is greater than or equal to the reliability polynomial of all other graphs with the same number of vertices and edges for all p ∈ [0, 1]. We examine the set of graphs with 8 vertices and 21 edges to determine if there exists a most reliable graph for this case. Specifically, we show that the graph whose compliment is 2K3 ∪ P2 is the uniformly most reliable graph with 8 vertices and 21 edges.

#### Zoom Link

https://uttyler.zoom.us/j/99124931036?pwd=YVYwME0vYkdPZi9GWVV3SnJ3cnJCZz09 (passcode: lyceum)

#### Keywords

Graph, Reliability, Polynomial

#### Persistent Identifier

http://hdl.handle.net/10950/3099

Finding a Uniformly Most Reliable Graph

The reliability polynomial of a simple graph G represents the probability that G will remain connected given a fixed probability 1 − p of each edge failing. A graph G is uniformly most reliable if its reliability polynomial is greater than or equal to the reliability polynomial of all other graphs with the same number of vertices and edges for all p ∈ [0, 1]. We examine the set of graphs with 8 vertices and 21 edges to determine if there exists a most reliable graph for this case. Specifically, we show that the graph whose compliment is 2K3 ∪ P2 is the uniformly most reliable graph with 8 vertices and 21 edges.