Presented by: Mujiangshan Wang

This report mainly focuses on the fault tolerance of networks (graphs), which is measured in two ways. The first is preclusion, which is a vertex set, whose removal leads to the destruction of all the subgraphs of specific size in the hierarchical network and the preclusion number is the minimum of this vertex set. Here we investigate a class of graphs which are constructed by combining the star graph with the bubble-sort graph, and give some preclusion numbers and edge preclusion numbers for this class of graphs. The second is $g$-good-neighbor diagnosability, which is the cardinality of a faulty vertex set, whose removal leads to that each vertex has at least $g$ fault-free neighbors in the remaining graph. Here we prove the $1$-good-neighbor diagnosability of $CK_n$ under the PMC model is $n^2-n-1$ for $n\geq 4$ and under the MM$^*$ model is $n^2-n-1$ for $n\geq 5$.

### About Pure mathematics seminars

We present regular seminars on a range of pure mathematics interests. Students, staff and visitors to UQ are welcome to attend, and to suggest speakers and topics.

Seminars are held on Tuesdays from 3pm to 3.50pm in Room 67-442 of the Priestley Building (Building 67).

Talks comprise 45 minutes of speaking time plus five minutes for questions and discussion.

#### Information for speakers

Researchers in all pure mathematics fields attend our seminars, so please aim your presentation at a general mathematical audience.

Room 67-442 has a data projector and a whiteboard.

If you wish to use the data projector, contact us a few days in advance of your talk to avoid technical delays on the day - there's a tight turnaround with room bookings.

#### Contact us

To arrange to use the projector, contact Benjamin Burton.

To volunteer to talk or to suggest a speaker, email Clement Maria or Huy Nguyen.