Leaky Positive Semidefinite Forcing on Graphs

We introduce $\mathrm{\ell }$-leaky positive semidefinite forcing and the $\mathrm{\ell }$-leaky positive semidefinite number of a graph, $Z_{(\mathrm{\ell })}^{+}G$, which combines the positive semidefinite color change rule with the addition of leaks to the graph. Furthermore, we determine general properties of $Z_{(\mathrm{\ell })}^{+}G$ and $Z_{(\mathrm{\ell })}^{+}G$ for various graphs, including path graphs, complete graphs, wheel graphs, complete bipartite graphs, trees, hypercubes, and prisms. We also define $\mathrm{\ell }$-leaky positive semidefinite forts with the purpose of unveiling differences between $\mathrm{\ell }$-leaky standard forcing and $\mathrm{\ell }$-leaky positive semidefinite forcing.