TITLE: Terminology in combinatorics
QUESTION [6 upvotes]: I met the following two combinatorial concepts during a study outside of combinatorics. I am wondering if there are common terminologies in combinatorics. 

A finite graph $G$ has the following property:


For any vertex $a$ and edge $\{b,c\}$ of $G$, there is an edge connecting them: there is at least one of $\{a,b\}$ or $\{a,c\}$ in $G$.


Let $P = \lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_k$ be an integer partition of $n$. Suppose that $P$ has the following property:


If there are two index sets $I$ and $J$ such that $\sum_{i \in I}\lambda_i = \sum_{j \in J}\lambda_j$, then the restricted partitions $\{\lambda_i\}_{i \in I}$ and $\{\lambda_j\}_{j \in J}$ are same. 

For instance, $5 \ge 3 \ge 2 \ge 2$ does not have the property because $2+3=5$, but $5 \ge 5 \ge 3 \ge 3$ has the property.
Are there common names for these two properties?

REPLY [2 votes]: The second property is almost exactly the concept of "distinct subset sums", about which there are many beautiful unanswered questions. See, eg, http://www.openproblemgarden.org/op/sets_with_distinct_subset_sums
The only difference between what you ask and the classical case is that you allow for repeated values.