Unicyclic SSP graphs of girth at least six

Determine whether any unicyclic graphs of girth g\geq 6 are SSP graphs, and classify which tadpole graphs T_{m,n} with m\geq 6 are SSP graphs.

Background

The paper shows that the tadpole graphs T_{4,n} and T_{5,n} are SSP graphs, but gives a counterexample demonstrating that T_{6,1} is not an SSP graph. It also notes that, to the authors’ knowledge, no unicyclic graph of girth at least six was known to belong to the class of SSP graphs. The unresolved issue is both whether such graphs exist at all and, specifically, which higher-girth tadpole graphs have the property.

References

To the best of our knowledge, there is no known unicyclic graph of girth at least $6$ in $$. Are there any unicyclic graphs of girth $g\geq 6$ in $$? Which tadpole graphs $T_{m,n}$, $m\geq 6$, are 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, Question immediately following the example showing T_{6,1} is not an SSP graph