Rooted product of graphs¶
The rooted product is a subgraph of the cartesian product of the same two graphs.
Core Idea¶
Rooted product of graphs is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: The rooted product is a subgraph of the cartesian product of the same two graphs. In mathematical graph theory, the rooted product (or comb product) of a graph and a rooted graph is defined as follows: take copies of , and for every vertex of , identify with the root node of the -th copy of . If is also rooted at , one can view the product itself as rooted, at .
Scope of Application¶
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Applications. (1980) used rooted products to find graceful numberings for a wide family of trees.
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Applications. These graphs can be used to generate examples in which the bound of Vizing's conjecture, an unproven inequality between the domination number of the graphs in a different graph product, the.
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More formally, assuming that. V = \left{(gi, hj): 1\leq i\leq n, 1\leq j\leq m\right}.
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More formally, assuming that. E = \Bigl{\bigl((gi, h1), (gk, h1)\bigr): (gi, gk) \in E(G)\Bigr} \cup \bigcup{i=1}^n \Bigl{\bigl((gi, hj), (gi, hk)\bigr): (hj, hk) \in.
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More formally, assuming that. If is also rooted at , one can view the product itself as rooted, at .
Clarity¶
A clear use of Rooted product of graphs names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is The rooted product is a subgraph of the cartesian product of the same two graphs. The strongest recognition evidence in the frozen account is: The rooted product is a subgraph of the cartesian product of the same two graphs.
Manages Complexity¶
Rooted product of graphs compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—e = \Bigl{\bigl((gi, h1), (gk, h1)\bigr): (gi, gk) \in E(G)\Bigr} \cup \bigcup{i=1}^n \Bigl{\bigl((gi, hj), (gi, hk)\bigr): (hj, hk) \in E(H)\Bigr} .—and the practical consequence—(1980) used rooted products to find graceful numberings for a wide family of trees.
Abstract Reasoning¶
- Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
- State the relation. Use the source-grounded identity: The rooted product is a subgraph of the cartesian product of the same two graphs.
- Check operation and conditions. If is also rooted at , one can view the product itself as rooted, at .
- Demand recognition evidence. The rooted product is a subgraph of the cartesian product of the same two graphs.
- Test variation.
Knowledge Transfer¶
Within the home domain. Knowledge about Rooted product of graphs transfers literally when a new case preserves the same carrier type, relation, and recognition test. (1980) used rooted products to find graceful numberings for a wide family of trees. These graphs can be used to generate examples in which the bound of Vizing's conjecture, an unproven inequality between the domination number of the graphs in a different graph product, the cartesian product of graphs, is exactly met . Beyond the home domain. No canonical parent is asserted for Rooted product of graphs.
Relationships to Other Abstractions¶
Current abstraction Rooted product of graphs Domain-specific
Parents (1) — more general patterns this builds on
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Rooted product of graphs is a kind of Network Prime
A rooted product of graphs is itself a graph/network constructed from rooted component graphs.
Hierarchy path (1) — routes to 1 parentless root
- Rooted product of graphs → Network → Reservoir-Flux Network → Conservation Laws → Invariance
Neighborhood in Abstraction Space¶
Rooted product of graphs sits in a crowded region of the domain-specific corpus (19th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Service-Quality Rates & Queueing Metrics (13 abstractions)
Nearest neighbors
- Typing Environment — 0.92
- Single Vegetative Obstruction Model — 0.91
- Filling radius — 0.91
- Julia set — 0.90
- Quasi-Frobenius Lie algebra — 0.90
Computed from structural-signature embeddings · 2026-10-08