資料結構與演算法

(5 points) Graph theory can be used to find the numbering of a chemical structure; a possible one can be used according to the following steps:

• a) Represent the molecule as an undirected graph $G = (V, E)$, where $V$ is the set of vertices and $E$ is the set of edges (bonds).
• b) Initialize each vertex $v \in V$ with a value equal to its degree (number of incident edges).
• c) For a fixed number of iterations or until convergence: (i) for each vertex $v$, compute a new value as the sum of the current values of its neighbors (not including itself); (ii) update all vertex values simultaneously.
• d) After the iterations, sort the vertices based on their final values in non-increasing order, ties breaking with smaller labels.

Consider the following 2-Carboxyoxybenzoic acid structure represented as a graph (numbers in parentheses are temporary labels for reference, not the final numbering). The graph has 13 vertices connected by the following bonds:

• Six-membered ring: $5\text{–}6,\ 6\text{–}7,\ 7\text{–}8,\ 8\text{–}9,\ 9\text{–}10,\ 10\text{–}5$
• Ester/acid branch on vertex 5: $5\text{–}4,\ 4\text{–}3,\ 3\text{–}1,\ 3\text{–}2$
• Carboxyl branch on vertex 10: $10\text{–}11,\ 11\text{–}12,\ 11\text{–}13$

(Vertices 1, 2, 12, 13 are oxygen leaves; 3 and 11 are the carboxyl/ester carbons; 4 is the bridging ester oxygen; 5–10 form the ring. Double bonds do not affect the simple-graph degree.)

Apply the above algorithm for two iterations on this graph. What is the correct ordering of the vertices after these iterations?