Computational methods for finding bi-regular cages

An $(\{r,m\};g)$-graph is a (simple, undirected) graph of girth $g\ge 3$ with vertices of degrees $r$ and $m$ where $2\le r<m$ . Given $r,m,g$, we seek the $(\{r,m\};g)$-graphs of minimum order, called $(\{r,m\};g)$-cages or bi-regular cages, whose order is denoted by $n(\{r,m\};g)$. In this paper, we use computational methods for finding $(\{r,m\};g)$-graphs of small order. Firstly, we present an exhaustive generation algorithm, which leads to $\text{unicode}x2013$ previously unknown $\text{unicode}x2013$ exhaustive lists of $(\{r,m\};g)$-cages for 24 different triples $(r,m,g)$. This also leads to the improvement of the lower bound of $n(\{4,5\};7)$ from 66 to 69. Secondly, we improve 49 upper bounds of $n(\{r,m\};g)$ based on constructions that start from $r$-regular graphs. Lastly, we generalize a theorem by Aguilar, Araujo-Pardo and Berman [arXiv:2305.03290, 2023], leading to 73 additional improved upper bounds.