Other girth-four and girth-five unicyclic SSP graphs

Determine whether any unicyclic graphs of girth four or five, other than the tadpole graphs T_{4,n} and T_{5,n}, are SSP graphs.

Background

The paper completely characterizes unicyclic SSP graphs of girth three and proves that every tadpole graph T_{4,n} and T_{5,n} is an SSP graph. However, it presents several counterexamples among other unicyclic graphs of girth four and five, including graphs with multiple degree-three vertices on the cycle and certain corona-type constructions. The authors therefore leave unresolved whether the two tadpole families exhaust the SSP unicyclic graphs in these two girths.

References

Are there any girth four or five unicyclic graphs except $T_{4,n}$ or $T_{5,n}$ in $$?

— The strong spectral property for some families of unicyclic graphs  (2501.01719 - Koljančić et al., 3 Jan 2025) in Section 6, Further examples and questions, immediately following the introductory paragraph