### Identifying Codes in Vertex-Transitive Graphs and Strongly Regular Graphs

#### Abstract

We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and $2\ln(|V|)+1$ where $V$ is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order $|V|^{\alpha}$ with $\alpha \in \{\frac{1}{4},\frac{1}{3},\frac{2}{5}\}$. These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs.

#### Keywords

Graph theory; identifying codes; metric-dimension; vertex-transitive graphs; strongly regular graphs; finite geometry; generalized quadrangles