$\mathrm{SDD}_1$ matrices and their generalizations

The paper considers the classes of $\mathrm{GSDD}_1$, $\mathrm{GSDD}_1^*$, and $SD$-$\mathrm{SDD}$ matrices, which contain the class of $\mathrm{SDD}$ (strictly diagonally dominant) matrices and are contained in the class of nonsingular $\mathcal{H}$-matrices. New upper bounds on $\|A^{-1}\|_\infty$ for $\mathrm{GSDD}_1$, $\mathrm{GSDD}_1^*$, and $SD$-$\mathrm{SDD}$ matrices $A$, generalizing known upper bounds for $S$-$\mathrm{SDD}$, $\mathrm{SDD}_1^*$, and $\mathrm{GSDD}_1$ matrices, are established and compared.