# Existence of d-regular nut graphs for d at most 11

A nut graph is a singular graph with one-dimensional kernel and corresponding eigenverctor with no 0 elements. The problem of determining the orders n for which d-regular nut graphs exist was recently posed by Gauci, Pisanski and Sciriha. These orders are known for d ≤ 4. Here we solve the problem for all remaining cases d≤ 11 and determine the complete lists of all d-regular nut graphs of order n for small values of d and n. The existence or non-existence of small regular nut graphs is determined by a computer search. The main tool is a construction that produces, for any d-regular nut graph of order n, another d-regular nut graph of order n + 2d. If we are given a sufficient number of d-regular nut graphs of consecutive orders, called seed graphs, this construction may be applied in such a way that the existence of all d-regular nut graphs of higher orders are established. For even d the orders n are indeed consecutive, while for odd d the orders n are consecutive even numbers. Furthermore, necessary conditions for combinations of order and degree for vertex-transitive nut graphs are derived.

• 3 publications
• 2 publications
• 22 publications
• 2 publications
• 4 publications
research
08/30/2019

### Existence of regular nut graphs for degree at most 11

A nut graph is a singular graph with one-dimensional kernel and correspo...
research
05/31/2022

### On the coset graph construction of distance-regular graphs

We show that no more new distance-regular graphs in the tables of the bo...
research
07/29/2019

### A new record of enumeration of regular graphs by parallel processing

We broke a record, set 8 years ago, in the enumeration of non-isomorphic...
research
08/22/2022

### Digraph Networks and Groupoids

We answer a question posed by Dougherty and Faber in [3], "Network routi...
research
08/20/2020

### On Fine-Grained Exact Computation in Regular Graphs

We show that there is no subexponential time algorithm for computing the...
research
08/26/2019

### Breadth-first search on a Ramanujan graph

The behavior of the randomized breadth-first search algorithm is analyze...
research
01/12/2023

### On Existence Theorems for Conditional Inferential Models

The framework of Inferential Models (IMs) has recently been developed in...