  The influence of node functionality (f) on the fracture of polymer networks
remains unclear. While many studies have focused on multi-functional nodes with
f>4, recent research suggests that networks with f=3 exhibit superior fracture
properties compared to those with f=4. To clarify this discrepancy, we
conducted phantom chain simulations for star-polymer networks varying f between
3 and 8. Our simulations utilized equimolar binary mixtures of star branch
prepolymers with a uniform arm length. We employed a Brownian dynamics scheme
to equilibrate sols and induce gelation through end-linking reactions. We
prevented the formation of odd-order loops owing to the binary reaction and
second-order loops algorithmically. We stored network structures at various
conversion ratios ({\phi}_c) and minimized energy to reduce computation costs
induced by structural relaxation. We subjected the networks to stretching until
fracture to determine stress and strain at break and work for fracture,
{\epsilon}_b, {\sigma}_b, and W_b. These fracture characteristics are highly
dependent on {\phi}_c for networks with small f but relatively insensitive for
those with large f. Thus, the networks with small f exhibit greater fracture
properties than those with large f at high {\phi}_c, whereas the opposite
relationship occurs at low {\phi}_c. We analyzed {\epsilon}_b, {\sigma}_b, and
W_b concerning the cycle rank {\xi} and the broken strand fraction {\phi}_bb.
We found {\epsilon}_b, {\sigma}_b/{\phi}_bb, and W_b/{\phi}_bb monotonically
decrease with increasing {\xi}, and the data for various f and {\phi}_c
superpose with each other to draw master curves. These results imply that the
mechanical superiority of the networks with small f comes from their smaller
{\xi} that gives higher {\epsilon}_b, {\sigma}_b/{\phi}_bb, and W_b/{\phi}_bb
than the networks with large f.
