The sign pattern of a real matrix $A$, denoted by $\mathrm{sgn}A$, is the $(+,-,0)$-matrix obtained from $A$ by replacing each entry by its sign. Let $𝒬\left(A\right)$ denote the set of all real matrices $B$ such that $\mathrm{sgn}B=\mathrm{sgn}A$. For a square real matrix $A$, the Drazin inverse of $A$ is the unique real matrix $X$ such that $A^{k+1}X=A^k$, $XAX=X$ and $AX=XA$, where $k$ is the Drazin index of $A$. We say that $A$ has signed Drazin inverse if $\mathrm{sgn}{\tilde{A}}^{\mathrm{d}}=\mathrm{sgn}{A}^{\mathrm{d}}$ for any $\tilde{A}\in 𝒬\left(A\right)$, where $A^{\mathrm{d}}$ denotes the Drazin inverse of $A$. In this paper, we give necessary conditions for some block triangular matrices to have signed...

The zero forcing number and the positive zero forcing number of a graph are two graph parameters that arise from two types of graph colourings. The zero forcing number is an upper bound on the minimum number of induced paths in the graph that cover all the vertices of the graph, while the positive zero forcing number is an upper bound on the minimum number of induced trees in the graph needed to cover all the vertices in the graph. We show that for a block-cycle graph the zero forcing number equals...