A good strategy to find examples that break this bound is to use Xi's construction of the dual extension algebra, which keeps the number of vertices, doubles the number of arrows, and exactly doubles the global dimension:

Xi, Changchang, Global dimensions of dual extension algebras, Manuscr. Math. 88, No. 1, 25-31 (1995). ZBL0851.16003.

For instance if the quiver is an oriented cycle ($n=m$), then the maximal finite global dimension that can be attained is $2n-2$, see

Gustafson, William H., Global dimension in serial rings, J. Algebra 97, 14-16 (1985). ZBL0571.16011.

Since $\mathrm{gldim} (A)=2n -2 < 2n=n_A+m_A$, such an algebra $A$ with maximal finite global dimension is not itself a counterexample. But for its dual extension algebra $D$, we get $\mathrm{gldim} (D)=4n -4$ and $n_D+m_D=n + 2n = 3n$. So for $n \geq 5$, $$\mathrm{gldim} (D) >n_D+m_D,$$ and the bound does not hold.

A good strategy to find examples that break this bound is to use Xi's construction of the dual extension algebra, which keeps the number of vertices, doubles the number of arrows, and exactly doubles the global dimension:

Xi, Changchang, Global dimensions of dual extension algebras, Manuscr. Math. 88, No. 1, 25-31 (1995). ZBL0851.16003.

For instance if the quiver is an oriented cycle ($n=m$), then the maximal finite global dimension that can be attained is $2n-2$, see

Gustafson, William H., Global dimension in serial rings, J. Algebra 97, 14-16 (1985). ZBL0571.16011.

Since $\mathrm{gldim} (A)=2n -2 < 2n=n_A+m_A$, such an algebra $A$ with maximal finite global dimension is not itself a counterexample. But for its dual extension algebra $D$, we get $\mathrm{gldim} (D)=4n -4$ and $n_D+m_D=n + 2n = 3n$. So for $n \geq 5$, $$\mathrm{gldim} (D) >n_D+m_D,$$ and the bound does not hold.

A good strategy to find examples that break this bound is to use Xi's construction of dual extension algebra, which keeps the number of vertices, doubles the number of arrows, and exactly doubles the global dimension:

Xi, Changchang, Global dimensions of dual extension algebras, Manuscr. Math. 88, No. 1, 25-31 (1995). ZBL0851.16003.

For instance if the quiver is an oriented cycle ($n=m$), then the maximal finite global dimension that can be attained is $2n-2$, see

Gustafson, William H., Global dimension in serial rings, J. Algebra 97, 14-16 (1985). ZBL0571.16011.

Since $\mathrm{gldim} (A)=2n -2 < 2n=n_A+m_A$, such an algebra $A$ is not itself a counterexample. But for its dual extension algebra $D$ we get $\mathrm{gldim} (D)=4n -4$ and $n_D+m_D=n + 2n = 3n$. So for $n > 5$, $$\mathrm{gldim} (D) >n_D+m_D,$$ and the bound does not hold.

A good strategy to find examples that break this bound is to use Xi's construction of the dual extension algebra, which keeps the number of vertices, doubles the number of arrows, and exactly doubles the global dimension:

Xi, Changchang, Global dimensions of dual extension algebras, Manuscr. Math. 88, No. 1, 25-31 (1995). ZBL0851.16003.

For instance if the quiver is an oriented cycle ($n=m$), then the maximal finite global dimension that can be attained is $2n-2$, see

Gustafson, William H., Global dimension in serial rings, J. Algebra 97, 14-16 (1985). ZBL0571.16011.

Since $\mathrm{gldim} (A)=2n -2 < 2n=n_A+m_A$, such an algebra $A$ with maximal finite global dimension is not itself a counterexample. But for its dual extension algebra $D$, we get $\mathrm{gldim} (D)=4n -4$ and $n_D+m_D=n + 2n = 3n$. So for $n \geq 5$, $$\mathrm{gldim} (D) >n_D+m_D,$$ and the bound does not hold.