Task: Neighbor

Problem:

Your task is to find the common neighbors of two nodes in an undirected academic network. In this network, nodes represent authors and edges represent research collaborations.

Example 1

Authors in the network: Keng Peng Tee, Toshiyuki Ohtsuka, David Q. Mayne, Darryl DeHaan, Miroslav Krstic, James B. Rawlings, Knut Graichen, Andreas Kugi, Pierre O. M. Scokaert, Christian Ebenbauer
Research collaborations between these authors: Keng Peng Tee and Miroslav Krstic, Keng Peng Tee and James B. Rawlings, Keng Peng Tee and David Q. Mayne, Keng Peng Tee and Pierre O. M. Scokaert, Keng Peng Tee and Darryl DeHaan, Toshiyuki Ohtsuka and Knut Graichen, Toshiyuki Ohtsuka and Miroslav Krstic, David Q. Mayne and Knut Graichen, David Q. Mayne and James B. Rawlings, David Q. Mayne and Christian Ebenbauer, David Q. Mayne and Pierre O. M. Scokaert, Darryl DeHaan and Knut Graichen, Darryl DeHaan and Christian Ebenbauer, Miroslav Krstic and Andreas Kugi, Miroslav Krstic and Christian Ebenbauer, James B. Rawlings and Knut Graichen, James B. Rawlings and Pierre O. M. Scokaert, James B. Rawlings and Christian Ebenbauer, Knut Graichen and Andreas Kugi, Knut Graichen and Pierre O. M. Scokaert, Pierre O. M. Scokaert and Christian Ebenbauer.

Common neighbors between Keng Peng Tee and Toshiyuki Ohtsuka: [Miroslav Krstic]

Problem to Solve

Authors in the network: Manfred Schmidt-Schauss, David Sabel, Markus Lohrey, Sebastian Maneth, Conrad Rau, Manfred Schmidt-Schauß, Jordi Levy
Research collaborations between these authors: Manfred Schmidt-Schauss and Jordi Levy, Manfred Schmidt-Schauss and David Sabel, Manfred Schmidt-Schauss and Manfred Schmidt-Schauß, Manfred Schmidt-Schauss and Markus Lohrey, Manfred Schmidt-Schauss and Sebastian Maneth, Manfred Schmidt-Schauss and Conrad Rau, David Sabel and Conrad Rau, David Sabel and Manfred Schmidt-Schauß, Markus Lohrey and Manfred Schmidt-Schauß, Markus Lohrey and Jordi Levy, Markus Lohrey and Sebastian Maneth, Sebastian Maneth and Manfred Schmidt-Schauß, Sebastian Maneth and Jordi Levy, Conrad Rau and Manfred Schmidt-Schauß, Manfred Schmidt-Schauß and Jordi Levy. Please identify the common neighbors of Manfred Schmidt-Schauss and David Sabel in this network. Present your answer in the following format: [AuthorA, AuthorB, AuthorC, AuthorD, ...].

Response:

Claude3-haiku:

[Manfred Schmidt-Schauß, Conrad Rau]

Deepseek-V2:

[Conrad Rau, Manfred Schmidt-Schauß, Markus Lohrey, Sebastian Maneth, Jordi Levy]

Gemma-7b:

[Conrad Rau, Manfred Schmidt-Schauss]

GPT-3.5:

[Manfred Schmidt-Schauß, Jordi Levy]

GPT-4o:

[Manfred Schmidt-Schauß, Conrad Rau]

Llama3-70b:

[Conrad Rau, Manfred Schmidt-Schauß]

Llama3-8b:

[Conrad Rau, Manfred Schmidt-Schauß]

Mixtral-7x8b:

[Manfred Schmidt-Schauß]

Qwen1.5-72b:

[Manfred Schmidt-Schauß]

Qwen1.5-7b:

[Manfred Schmidt-Schauß, David Sabel, Markus Lohrey]

Task: Component

Problem:

You are required to identify all connected components in the given social network and output one representative node from each component. Within a connected component, any node can be reached from any other node through the edges in the graph. Different connected components are isolated from each other.

Example 1

Names in the network: Barbara Perkins, Jonathon Mitchell, Jenny Trujillo, Adam Hoffman, Devin Nunez.
Friendship connections: Barbara Perkins and Jonathon Mitchell, Jenny Trujillo and Devin Nunez, Jenny Trujillo and Adam Hoffman, Adam Hoffman and Devin Nunez.

The answer including one representative element from each connected component in the given social network: [Barbara Perkins, Jenny Trujillo]

Problem to Solve

Names in the network: Veronica Garcia, Katherine Brennan, Angel Chavez, Steven Martin, Brett Johnson, Megan Banks, Julia Dominguez, Rachel Mitchell
Fiendship connections: Veronica Garcia to Brett Johnson, Veronica Garcia to Megan Banks, Katherine Brennan to Brett Johnson, Katherine Brennan to Megan Banks, Angel Chavez to Megan Banks, Angel Chavez to Rachel Mitchell, Steven Martin to Megan Banks, Brett Johnson to Megan Banks, Megan Banks to Julia Dominguez, Megan Banks to Rachel Mitchell Identify all connected components in this network. Note that for each connected component, you should only output one of its nodes. Present your answer in the following format: [UserA, UserB, UserC, UserD, ...]

Response:

Claude3-haiku:

[Veronica Garcia, Katherine Brennan, Angel Chavez, Steven Martin, Julia Dominguez]

Deepseek-V2:

[Veronica Garcia, Angel Chavez, Steven Martin]

Gemma-7b:

[Veronica Garcia, Katherine Brennan, Angel Chavez, Steven Martin, Brett Johnson, Megan Banks, Julia Dominguez, Rachel Mitchell]

GPT-3.5:

[Veronica Garcia, Katherine Brennan, Angel Chavez, Steven Martin, Megan Banks, Julia Dominguez, Rachel Mitchell]

GPT-4o:

[Veronica Garcia, Angel Chavez, Steven Martin]

Llama3-70b:

[Veronica Garcia, Angel Chavez]

Llama3-8b:

[Veronica Garcia, Megan Banks, Rachel Mitchell]

Mixtral-7x8b:

In this social network, there are two connected components. The first connected component consists of the nodes Veronica Garcia, Brett Johnson, and Megan Banks. The second connected component includes Katherine Brennan, Angel Chavez, Rachel Mitchell, Julia Dominguez.

The representative nodes from each connected component are Veronica Garcia and Katherine Brennan.

Therefore, the answer is: [Veronica Garcia, Katherine Brennan]

Qwen1.5-72b:

[Veronica Garcia, Katherine Brennan, Angel Chavez, Steven Martin, Brett Johnson, Megan Banks, Julia Dominguez, Rachel Mitchell]

Qwen1.5-7b:

[Veronica Garcia, Katherine Brennan, Angel Chavez, Steven Martin]

Task: Diameter

Problem:

You are required to calculate the diameter of an undirected knowledge graph. The diameter of a graph is the maximum distance between any pair of nodes in the graph. To compute this, you need to find the shortest path between all pairs of nodes and then determine the maximum length of these shortest paths.

Example 1

Entities in this knowledge graph: Village, Niwica Żary County, Prosity, Warmian-Masurian Voivodeship, Dębinka Lubusz Voivodeship, Gmina Trzebiel, Święta Lipka, Poland, Stare Leśno, Wielki Łęck, Gorzów County.
The relationships between these entities are as follows:
Village is connected to Wielki Łęck via the relationship type.
Village is connected to Dębinka Lubusz Voivodeship via the relationship type.
Village is connected to Niwica Żary County via the relationship type.
Village is connected to Prosity via the relationship type.
Village is connected to Święta Lipka via the relationship type.
Niwica Żary County is connected to Gorzów County via the relationship http://www.w3.org/2000/01/rdf-schema#seeAlso.
Niwica Żary County is connected to Poland via the relationship country.
Niwica Żary County is connected to Gmina Trzebiel via the relationship isPartOf.
Prosity is connected to Poland via the relationship country.
Prosity is connected to Warmian-Masurian Voivodeship via the relationship isPartOf.
Warmian-Masurian Voivodeship is connected to Wielki Łęck via the relationship isPartOf.
Warmian-Masurian Voivodeship is connected to Dębinka Lubusz Voivodeship via the relationship http://www.w3.org/2000/01/rdf-schema#seeAlso.
Warmian-Masurian Voivodeship is connected to Poland via the relationship country.
Warmian-Masurian Voivodeship is connected to Święta Lipka via the relationship isPartOf.
Dębinka Lubusz Voivodeship is connected to Poland via the relationship country.
Dębinka Lubusz Voivodeship is connected to Gmina Trzebiel via the relationship isPartOf.
Gmina Trzebiel is connected to Poland via the relationship country.
Święta Lipka is connected to Poland via the relationship country.
Poland is connected to Gorzów County via the relationship country.
Poland is connected to Stare Leśno via the relationship country.
Poland is connected to Wielki Łęck via the relationship country.

One shortest path corresponding to the diameter in this graph is: [Village, Święta Lipka, Poland, Stare Leśno]

Problem to Solve

Entities in this knowledge graph: Seine-et-Marne, Montereau-Fault-Yonne, Vittorio De Sica, Saint-Fiacre Seine-et-Marne, France
The relationships between these entities are as follows:
Seine-et-Marne is connected to Saint-Fiacre Seine-et-Marne via the relationship department.
Seine-et-Marne is connected to Montereau-Fault-Yonne via the relationship department.
Seine-et-Marne is connected to France via the relationship country.
Montereau-Fault-Yonne is connected to France via the relationship country.
Vittorio De Sica is connected to France via the relationship deathPlace.
Saint-Fiacre Seine-et-Marne is connected to France via the relationship country. Please determine the diameter of this network and output the corresponding path in the following format: [Entity1, Entity2, ..., EntityN].

Response:

Claude3-haiku:

[Seine-et-Marne, Saint-Fiacre Seine-et-Marne, France, Montereau-Fault-Yonne]

Deepseek-V2:

To calculate the diameter of the given knowledge graph, we need to find the longest shortest path between any two nodes. The graph consists of the following entities and relationships:

Seine-et-Marne is connected to Saint-Fiacre Seine-et-Marne via the relationship department.
Seine-et-Marne is connected to Montereau-Fault-Yonne via the relationship department.
Seine-et-Marne is connected to France via the relationship country.
Montereau-Fault-Yonne is connected to France via the relationship country.
Vittorio De Sica is connected to France via the relationship deathPlace.
Saint-Fiacre Seine-et-Marne is connected to France via the relationship country.

From these relationships, we can deduce the following paths:

Seine-et-Marne to Saint-Fiacre Seine-et-Marne (1 step)
Seine-et-Marne to Montereau-Fault-Yonne (1 step)
Seine-et-Marne to France (1 step)
Montereau-Fault-Yonne to France (1 step)
Vittorio De Sica to France (1 step)
Saint-Fiacre Seine-et-Marne to France (1 step)

The longest shortest path in this graph is 1 step, as all entities are directly connected to each other or to France. Therefore, the diameter of the graph is 1.

The corresponding path for the diameter is: [Seine-et-Marne, France]

Gemma-7b:

[Seine-et-Marne, Saint-Fiacre Seine-et-Marne, France]

GPT-3.5:

[Seine-et-Marne, France, Vittorio De Sica]

GPT-4o:

The given knowledge graph consists of the following entities and relationships:

Entities: Seine-et-Marne, Montereau-Fault-Yonne, Vittorio De Sica, Saint-Fiacre Seine-et-Marne, France
Relationships:
- Seine-et-Marne is connected to Saint-Fiacre Seine-et-Marne (department).
- Seine-et-Marne is connected to Montereau-Fault-Yonne (department).
- Seine-et-Marne is connected to France (country).
- Montereau-Fault-Yonne is connected to France (country).
- Vittorio De Sica is connected to France (deathPlace).
- Saint-Fiacre Seine-et-Marne is connected to France (country).

First, let's visualize the graph:

Seine-et-Marne is directly connected to:
- Saint-Fiacre Seine-et-Marne
- Montereau-Fault-Yonne
- France
Montereau-Fault-Yonne is directly connected to:
- Seine-et-Marne
- France
Saint-Fiacre Seine-et-Marne is directly connected to:
- Seine-et-Marne
- France
Vittorio De Sica is directly connected to:
- France
France is directly connected to:
- Seine-et-Marne
- Montereau-Fault-Yonne
- Saint-Fiacre Seine-et-Marne
- Vittorio De Sica

Now, we will determine the shortest paths between all pairs of nodes and find the longest of these shortest paths to determine the diameter of the graph.

The graph can be represented as follows:

Seine-et-Marne ↔ Saint-Fiacre Seine-et-Marne
Seine-et-Marne ↔ Montereau-Fault-Yonne
Seine-et-Marne ↔ France
Montereau-Fault-Yonne ↔ France
Saint-Fiacre Seine-et-Marne ↔ France
Vittorio De Sica ↔ France

Given the connections, the shortest paths between nodes are:

Seine-et-Marne to Saint-Fiacre Seine-et-Marne: 1 edge
Seine-et-Marne to Montereau-Fault-Yonne: 1 edge
Seine-et-Marne to France: 1 edge
Montereau-Fault-Yonne to France: 1 edge
Saint-Fiacre Seine-et-Marne to France: 1 edge
Vittorio De Sica to France: 1 edge

By analyzing the shortest paths, it is clear that the longest shortest path is between Vittorio De Sica and any other node except France, which is 2 edges long, such as:

[Vittorio De Sica, France, Montereau-Fault-Yonne]

Thus, the diameter of the graph is [2], and one of the paths corresponding to the diameter is:

[Vittorio De Sica, France, Montereau-Fault-Yonne]

Llama3-70b:

[Seine-et-Marne, Montereau-Fault-Yonne, France, Vittorio De Sica]

Llama3-8b:

To calculate the diameter of the graph, we need to find the shortest path between all pairs of nodes and then determine the maximum length of these shortest paths.

After analyzing the relationships between the entities in the Seine-et-Marne graph, we can identify the shortest paths between all pairs of nodes:

Seine-et-Marne to Saint-Fiacre Seine-et-Marne: [Seine-et-Marne, Saint-Fiacre Seine-et-Marne] (distance 1)
Seine-et-Marne to Montereau-Fault-Yonne: [Seine-et-Marne, Montereau-Fault-Yonne] (distance 1)
Seine-et-Marne to France: [Seine-et-Marne, France] (distance 1)
Montereau-Fault-Yonne to France: [Montereau-Fault-Yonne, France] (distance 1)
Vittorio De Sica to France: [Vittorio De Sica, France] (distance 1)
Saint-Fiacre Seine-et-Marne to France: [Saint-Fiacre Seine-et-Marne, France] (distance 1)

The maximum distance between any pair of nodes is 1, which means the diameter of the graph is 1.

The corresponding path for the diameter is: [Seine-et-Marne, Saint-Fiacre Seine-et-Marne, France]

Mixtral-7x8b:

The diameter of this network is 3, and the corresponding path is: [Seine-et-Marne, Montereau-Fault-Yonne, France, Vittorio De Sica]

Qwen1.5-72b:

The diameter of this network is 3. One possible shortest path corresponding to the diameter is: [Seine-et-Marne, Vittorio De Sica, France, Saint-Fiacre Seine-et-Marne]

Qwen1.5-7b:

The diameter of this knowledge graph can be calculated by finding the shortest path between all pairs of entities and then finding the maximum length. Here's the path with the maximum length:

[Vittorio De Sica, France, Seine-et-Marne, Saint-Fiacre Seine-et-Marne, France, Montereau-Fault-Yonne, France]

This path corresponds to the longest distance between any pair of entities in the graph. Therefore, the diameter is 6. The output format would be:

[Vittorio De Sica, France, Seine-et-Marne, Saint-Fiacre Seine-et-Marne, France, Montereau-Fault-Yonne, France]

Task: Distance

Problem:

Your task is to identify the shortest path between two specified entities in an undirected knowledge graph, minimizing the number of hops.

Example 1

Entities in this knowledge graph: Poland, Village, Warmian-Masurian Voivodeship, Dębinka Lubusz Voivodeship
The relationships between these entities are as follows:
Poland is connected to Warmian-Masurian Voivodeship via the relationship country.
Poland is connected to Dębinka Lubusz Voivodeship via the relationship country.
Village is connected to Dębinka Lubusz Voivodeship via the relationship type.
Warmian-Masurian Voivodeship is connected to Dębinka Lubusz Voivodeship via the relationship http://www.w3.org/2000/01/rdf-schema#seeAlso.

One shortest path between Dębinka Lubusz Voivodeship and Poland is: [Dębinka Lubusz Voivodeship, Poland]

Problem to Solve

Entities in this knowledge graph: Senegal, Government of Australia, Niger, Bamako, Greenwich Mean Time, Midfielder (association football), Mali, Burkina Faso, Mali national football team, Defender (association football), Ségou, Ouattagouna, Ansongo Cercle, Araouane, Mopti, Suðuroy, Midfielder
The relationships between these entities are as follows:
Senegal is connected to Greenwich Mean Time via the relationship timeZone.
Bamako is connected to Mali via the relationship capital.
Greenwich Mean Time is connected to Suðuroy via the relationship timeZone.
Greenwich Mean Time is connected to Ansongo Cercle via the relationship timeZone.
Greenwich Mean Time is connected to Ségou via the relationship timeZone.
Greenwich Mean Time is connected to Mali via the relationship timeZone.
Mali is connected to Ouattagouna via the relationship country.
Mali is connected to Ségou via the relationship country.
Mali is connected to Ansongo Cercle via the relationship country.
Mali is connected to Mopti via the relationship country.
Mali is connected to Araouane via the relationship country.
Ouattagouna is connected to Ansongo Cercle via the relationship isPartOf. Please determine the shortest path between Bamako and Suðuroy in this network. Submit your answer in the format: [Entity1, Entity2, ..., EntityN], where Entity1 and EntityN are the specified start and end entities, and Entity2 through EntityN-1 are the intermediate entities on the shortest path.

Response:

Claude3-haiku:

[Bamako, Mali, Greenwich Mean Time, Suðuroy]

Deepseek-V2:

[Bamako, Mali, Greenwich Mean Time, Suðuroy]

Gemma-7b:

[Bamako, Mali, Ansongo Cercle, Suðuroy]

GPT-3.5:

[ Bamako, Mali, Greenwich Mean Time, Suðuroy]

GPT-4o:

The shortest path between Bamako and Suðuroy in the given knowledge graph is: [Bamako, Mali, Greenwich Mean Time, Suðuroy].

Llama3-70b:

[Bamako, Mali, Greenwich Mean Time, Suðuroy]

Llama3-8b:

[Bamako, Mali, Greenwich Mean Time, Suðuroy]

Mixtral-7x8b:

[Bamako, Mali, Suðuroy]

Qwen1.5-72b:

The shortest path between Bamako and Suðuroy in this network is: [Bamako, Mali, Greenwich Mean Time, Suðuroy]

Qwen1.5-7b:

The shortest path between Bamako and Suðuroy in this network is: [Bamako, Mali, Greenwich Mean Time, Suðuroy]

Task: GED

Problem:

You are required to solve the Graph Edit Distance problem between two molecules. Each edit operation (adding or deleting an edge, adding or deleting an isolated node, or relabeling a node) has the identity cost. Your objective is to establish a mapping between the atom IDs from Molecule A to Molecule B, ensuring that each atom ID in Molecule A corresponds to exactly one atom ID in Molecule B. The mapping corresponds to the minimum edit cost between the two graphs.

Example 1

Molecule A:

Atoms: N (atom 0), O (atom 1), Si (atom 2), O (atom 3), O (atom 4).
Bonds: 0-1, 1-2, 2-3, 2-4. Molecule B:
Atoms: F (atom 0), B (atom 1), C (atom 2), N (atom 3), Br (atom 4).
Bonds: 0-1, 1-2, 1-4, 2-3.

One optimal node mapping: [3, 2, 1, 0, 4].

Problem to Solve

You are given the following two molecules: Molecule A:

Atoms: N (atom 0), C (atom 1), N (atom 2), F (atom 3).
Bonds: 0-1, 1-2, 1-3. Molecule B:
Atoms: O (atom 0), C (atom 1), F (atom 2), F (atom 3).
Bonds: 0-1, 1-2, 1-3. Represent the node mapping as a list of integers, where the position in the list corresponds to the atom ID in Molecule A and the value at that position indicates the corresponding atom ID in Molecule B. For instance, if atom 0 in Molecule A corresponds to atom 1 in Molecule B, atom 1 in Molecule A corresponds to atom 0 in Molecule B, and atom 2 remains unchanged, the mapping would be represented as [1, 0, 2, ...].

Response:

Claude3-haiku:

[2, 1, 0, 3]

Deepseek-V2:

To solve the Graph Edit Distance problem for the given molecules, we need to find the optimal mapping that minimizes the edit cost. Here are the steps to find the optimal mapping:

Identify the atoms and their labels in both molecules:
- Molecule A: N (atom 0), C (atom 1), N (atom 2), F (atom 3)
- Molecule B: O (atom 0), C (atom 1), F (atom 2), F (atom 3)
Analyze the bonds in both molecules:
- Molecule A: 0-1, 1-2, 1-3
- Molecule B: 0-1, 1-2, 1-3
Find the optimal mapping:
- Atom 0 in Molecule A (N) can map to atom 1 in Molecule B (C) or atom 0 in Molecule B (O). Mapping to atom 1 (C) aligns better with the bond structure.
- Atom 1 in Molecule A (C) can map to atom 0 in Molecule B (C) or atom 1 in Molecule B (C). Mapping to atom 0 (C) aligns with the bond structure.
- Atom 2 in Molecule A (N) can map to atom 2 in Molecule B (F) or atom 3 in Molecule B (F). Mapping to atom 2 (F) aligns with the bond structure.
- Atom 3 in Molecule A (F) can map to atom 2 in Molecule B (F) or atom 3 in Molecule B (F). Mapping to atom 3 (F) aligns with the bond structure.
Final mapping:
- The optimal mapping that maintains the bond structure and minimizes the edit cost is: [1, 0, 2, 3]

This mapping ensures that each atom ID in Molecule A corresponds to exactly one atom ID in Molecule B, and the bond structure is preserved as much as possible, minimizing the edit cost.

Gemma-7b:

[1, 0, 2, 3]

GPT-3.5:

The graph edit distance problem can be solved using the Hungarian algorithm to find the optimal node mapping between the two molecules. First, construct the cost matrix where the cost of matching atom i in Molecule A with atom j in Molecule B is the edit distance between the two atoms. Then, use the Hungarian algorithm to find the optimal assignment that minimizes the total cost.

For the given molecules: Molecule A:

Atoms: N (atom 0), C (atom 1), N (atom 2), F (atom 3).
Bonds: 0-1, 1-2, 1-3. Molecule B:
Atoms: O (atom 0), C (atom 1), F (atom 2), F (atom 3).
Bonds: 0-1, 1-2, 1-3.

The cost matrix is:

| 1 0 1 1 |
| 0 1 0 0 |
| 1 0 1 1 |
| 1 0 1 1 |

Using the Hungarian algorithm, the optimal node mapping is: [1, 2, 0, 3]

GPT-4o:

For the given problem of finding the Graph Edit Distance between two molecules with the provided data:

Molecule A:

Atoms: N (atom 0), C (atom 1), N (atom 2), F (atom 3)
Bonds: 0-1, 1-2, 1-3

Molecule B:

Atoms: O (atom 0), C (atom 1), F (atom 2), F (atom 3)
Bonds: 0-1, 1-2, 1-3

We need to find the optimal mapping of atoms from Molecule A to Molecule B that minimizes the edit distance. Considering the costs and the nature of atoms and bonds, the optimal mapping that aligns the most similar atoms and bond structures would be as follows:

[0, 1, 3, 2]

This means:

Atom 0 in Molecule A (N) maps to atom 0 in Molecule B (O)
Atom 1 in Molecule A (C) maps to atom 1 in Molecule B (C)
Atom 2 in Molecule A (N) maps to atom 3 in Molecule B (F)
Atom 3 in Molecule A (F) maps to atom 2 in Molecule B (F)

So the optimal node mapping is [0, 1, 3, 2]

Llama3-70b:

[0, 1, 2, 3]

Llama3-8b:

[Molecule A: N (atom 0), C (atom 1), N (atom 2), F (atom 3)] [Molecule B: O (atom 0), C (atom 1), F (atom 2), F (atom 3)]

One optimal node mapping: [1, 1, 2, 2].

Mixtral-7x8b:

The node mapping between Molecule A and Molecule B that minimizes the Graph Edit Distance is [1, 0, 3, 2]. Here's the reasoning:

Atom 0 (N) in Molecule A corresponds to Atom 1 (C) in Molecule B.
Atom 1 (C) in Molecule A corresponds to Atom 0 (O) in Molecule B.
Atom 2 (N) in Molecule A corresponds to Atom 3 (F) in Molecule B.
Atom 3 (F) in Molecule A remains unchanged, as there is no better match in Molecule B.

Qwen1.5-72b:

An optimal node mapping for the given molecules is [2, 1, 0, 3].

Qwen1.5-7b:

The node mapping for the given example is [1, 0, 2, 3].

Task: MCP

Problem:

You are required to solve the Maximum Clique Problem for an undirected academic network. In this network, nodes represent authors and edges represent research collaborations. Your objective is to find the largest subset of nodes such that every pair of vertices in this subset is connected by an edge.

Example 1

Authors in the network: Keng Peng Tee, Veit Hagenmeyer, Bartosz Käpernick, Karl Henrik Johansson, Darryl DeHaan, James B. Rawlings, Andreas Kugi, Knut Graichen, Tilman Utz, Christian Ebenbauer.
Research collaborations between these authors: Keng Peng Tee and James B. Rawlings, Keng Peng Tee and Darryl DeHaan, Veit Hagenmeyer and Knut Graichen, Bartosz Käpernick and Andreas Kugi, Bartosz Käpernick and Knut Graichen, Karl Henrik Johansson and Christian Ebenbauer, Darryl DeHaan and Knut Graichen, Darryl DeHaan and Christian Ebenbauer, James B. Rawlings and Knut Graichen, James B. Rawlings and Christian Ebenbauer, Andreas Kugi and Knut Graichen, Knut Graichen and Tilman Utz.

One Maximum Clique: [Knut Graichen, Bartosz Käpernick, Andreas Kugi].

Problem to Solve

Authors in the network: Manfred Schmidt-Schauss, David Sabel, Manfred Schmidt-Schauß, Guillem Godoy
Research collaborations between these authors: Manfred Schmidt-Schauss and David Sabel, Manfred Schmidt-Schauss and Manfred Schmidt-Schauß, Manfred Schmidt-Schauss and Guillem Godoy, David Sabel and Manfred Schmidt-Schauß, Manfred Schmidt-Schauß and Guillem Godoy. Identify the clique with the maximum number of authors in this network. Present your answer in the following format: [AuthorA, AuthorB, AuthorC, AuthorD, ...].