The organization of the paper is as follows. In section 1 we prove that a minimal-ABC tree with B1-branch of order 2 contains neither 4 nor 3 B1-branches. In section 2, we prove that the ABC index of a minimal-ABC tree with B1-branch of order 2 is at least 3. In section 3, we provide the experimental data and analyze the minimal-ABC trees with B1-branch of order 2. In section 4, the minimum number of B1-branches which are needed to minimize the ABC index is proven. In section 5, we show the results of the search for the minimal-ABC trees with B1-branch of order 2.
Thus, the LSM images have revealed the crack behaviors of the Ti film on the PDMS substrates at the micro- and nanoscales. The LSM images are found to be consistent with the optical microscopy images, as shown in Figure 2g,h. From the results obtained so far, it is found that the crack formation mainly occurs during the deformation of the PDMS substrate, and that the crack behaviors are strain dependent. The crack density is larger in the undeformed areas and becomes smaller as the strain becomes larger.
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