computer science

Hamiltonian cycle polynomial

Hamiltonian cycle polynomial is a computer science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Hamiltonian cycle polynomial rather than just read about it. In short: In mathematics, the Hamiltonian cycle polynomial of an n×n-matrix is a polynomial in its entries, defined as ham ⁡ ( A ) = ∑ σ ∈ H n ∏ i = 1 n a i , σ ( i ) {\displaystyle \operatorname {ham} (A)=\sum _{\sigma \in H_{n}}\prod _{i=1}^{n}a_{i,\sigma (i)}} where H n {\displaystyle H_{n}} is the set of n-permutations having exactly one cycle. This is an algebraic option useful, in a number of cases, for determining the…

Key takeaways

Hamiltonian cycle polynomial belongs to computer science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Hamiltonian cycle polynomial to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hamiltonian cycle polynomial from memory before moving on to harder problems.

Reference excerpt

In mathematics, the Hamiltonian cycle polynomial of an n×n-matrix is a polynomial in its entries, defined as

ham ⁡ ( A ) = ∑ σ ∈ H n ∏ i = 1 n a i , σ ( i ) {\displaystyle \operatorname {ham} (A)=\sum _{\sigma \in H_{n}}\prod _{i=1}^{n}a_{i,\sigma (i)}}

where H n {\displaystyle H_{n}} is the set of n-permutations having exactly one cycle. This is an algebraic option useful, in a number of cases, for determining the existence of a Hamiltonian cycle in a directed graph. It is a generalization of the number of Hamiltonian cycles of a digraph as the sum of the products of its Hamiltonian cycles' arc weights (all of which equal unity) for weighted digraphs with arc weights taken from a given commutative ring. In the meantime, for an undirected weighted graph the sum of the products of the edge weights of its Hamiltonian cycles containing any fixed edge (i,j) can be expressed as the product of the weight of (i,j) and the Hamiltonian cycle polynomial of a matrix received from its weighted adjacency matrix via subjecting its rows and columns to any permutation mapping i to 1 and j to 2 and then removing its 1-st row and 2-nd column. In (Knezevic & Cohen (2017)) it was shown that

ham ⁡ ( A ) = ∑ J ⊆ { 2 , … , n } det ( − A J ) per ⁡ ( A J ¯ ) {\displaystyle \operatorname {ham} (A)=\sum _{J\subseteq \{2,\dots ,n\}}\det(-A_{J})\operatorname {per} (A_{\bar {J}})}

where A J {\displaystyle A_{J}} is the submatrix of A {\displaystyle A} induced by the rows and columns of A {\displaystyle A} indexed by J {\displaystyle J} , and J ¯ {\displaystyle {\bar {J}}} is the complement of J {\displaystyle J} in { 1 , … , n } {\displaystyle \{1,\dots ,n\}} , while the determinant of the empty submatrix is defined to be 1. Due to this and Borchardt's identities, for a non-singular n×n Cauchy matrix C ( x , y ) {\displaystyle C(x,y)} ham ⁡ ( C ( x , y ) ) = det ( − D 1 2 C ∗ 2 ( x , y ) D 2 2 + I / 1 ) det ⁡ ( C ( x , y ) ) {\displaystyle \operatorname {ham} (C(x,y))={\det }(-D_{1}^{2}C^{*2}(x,y)D_{2}^{2}+I_{/1})\operatorname {det} (C(x,y))} where D 1 , D 2 {\displaystyle D_{1},D_{2}} are diagonal matrices that make D 1 C ( x , y ) D 2 {\displaystyle D_{1}C(x,y)D_{2}} unitary (in a real field or a field of a finite characteristic, or orthogonal in the field of complex numbers), C ∗ 2 ( x , y ) {\displaystyle C^{*2}(x,y)} is the Hadamard (entry-wise) square of C ( x , y ) {\displaystyle C(x,y)} , and I / 1 {\displaystyle I_{/1}} is the identity n×n-matrix with the entry of indexes 1,1 replaced by 0.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Hamiltonian cycle polynomial

Start with the simplest possible case. Write down what Hamiltonian cycle polynomial claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Hamiltonian cycle polynomial before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Hamiltonian cycle polynomial ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Hamiltonian cycle polynomial

In research: Hamiltonian cycle polynomial appears in computer science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Hamiltonian cycle polynomial in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Hamiltonian cycle polynomial is common in secondary-school and first-year university syllabi. It links to neighbouring topics Computational problems in graph theory, Hamiltonian paths and cycles, NP-complete problems, so understanding it makes those chapters shorter.
In everyday life: Look for Hamiltonian cycle polynomial outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Hamiltonian cycle polynomial in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hamiltonian cycle polynomial means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Hamiltonian cycle polynomial out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hamiltonian cycle polynomial in simple terms?

In mathematics, the Hamiltonian cycle polynomial of an n×n-matrix is a polynomial in its entries, defined as ham ⁡ ( A ) = ∑ σ ∈ H n ∏ i = 1 n a i , σ ( i ) {\displaystyle \operatorname {ham} (A)=\sum _{\sigma \in H_{n}}\prod _{i=1}^{n}a_{i,\sigma (i)}} where H n {\displaystyle H_{n}} is the set of…

Why does Hamiltonian cycle polynomial matter?

Because it connects several computer science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Hamiltonian cycle polynomial?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Hamiltonian cycle polynomial.