Questions: Graph Operations and Products

Question 1

In the Cartesian product G □ H, when are two vertices (u, a) and (v, b) adjacent?

• A. When u = v and a = b (they share both coordinates)
• B. When u–v is an edge in G and a–b is an edge in H (both coordinates change via valid edges)
• C. When exactly one coordinate changes and that change is a valid edge in the corresponding graph
• D. When at least one coordinate is the same regardless of whether the other forms a valid edge

Answer: (C)

The Cartesian product connects vertices that differ in exactly one coordinate, and that difference must be a valid edge in the corresponding graph. Option B describes the tensor product — the most common confusion between these two products. The 'exactly one coordinate' rule is what gives the Cartesian product its grid-like structure, analogous to adjacency in a rectangular grid.

Question 2

The tensor product G × H of two graphs produces a sparser graph than the Cartesian product G □ H on the same pair of graphs. Why?

• A. Because the tensor product uses a smaller vertex set than the Cartesian product
• B. Because the tensor product requires both coordinate changes to be valid edges simultaneously, creating fewer qualifying pairs
• C. Because the tensor product excludes vertices where both coordinates are identical
• D. Because the Cartesian product allows edges between any vertex pair while the tensor product restricts to graph edges

Answer: (B)

Both products have identical vertex sets — all pairs from V(G) × V(H). The difference is purely the edge rule. The tensor product requires a valid edge in G AND a valid edge in H simultaneously; both coordinates must advance together. This double requirement is harder to satisfy than the Cartesian product's 'exactly one coordinate' rule, so fewer vertex pairs qualify and the resulting graph is sparser.

Question 3

The Cartesian product and tensor product of two graphs always have the same vertex sets but different edge sets.

• T. True
• F. False

Answer: True

All graph products on graphs G and H use the same vertex set: V(G) × V(H) — one vertex for every pairing of a vertex from G with a vertex from H. What distinguishes products is purely the adjacency rule — which of those pairs are connected by edges. This is why comparing products requires understanding their edge rules, not their vertex constructions.

Question 4

The strong product G ⊠ H is defined by the same edge rule as the tensor product but applied to a larger vertex set.

• T. True
• F. False

Answer: False

The strong product uses the same vertex set as all other products on G and H. What makes it 'strong' is its edge rule: two vertices are adjacent if they are adjacent in either the Cartesian product OR the tensor product — it is the union of both products' edge sets. This makes the strong product the densest of the three. The vertex set plays no role in the distinction.

Question 5

All three major graph products (Cartesian, tensor, strong) are defined on the same vertex set. What varies between them, and what is the significance of having different edge rules?

Model answer: What varies is the adjacency rule — which pairs of vertices in V(G) × V(H) are connected by edges. Cartesian product: exactly one coordinate changes via a valid edge. Tensor product: both coordinates simultaneously change via valid edges. Strong product: union of both (adjacent in either). Different edge rules produce graphs with different densities and structural properties, making each product appropriate for different applications — grid networks, graph homomorphisms, or multi-dimensional relationship structures.

The key insight is that all graph products share the same combinatorial vertex space; what distinguishes them is how 'movement' through the two underlying graphs is coordinated. The Cartesian product allows movement in one dimension at a time; the tensor product requires simultaneous movement in both; the strong product allows either.