Intereting Posts

Determine cycle from adjacency matrix
Finding the boundaries on $\rho$ in spherical coordinates
Monkey typing ABRACADABRA and gamblers
Power Series with the coefficients $n!/(n^n)$
The Integral that Stumped Feynman?
The continuity of multivariable function
A function for which the Newton-Raphson method slowly converges?
Hatcher question: How to Cut and Glue from Tetrahedron to Klein Bottle
Analytic continuation of factorial function
Three negative and three positive $1$s in a serie
Integral Inequality Absolute Value: $\left| \int_{a}^{b} f(x) g(x) \ dx \right| \leq \int_{a}^{b} |f(x)|\cdot |g(x)| \ dx$
Prove that the sphere is the only closed surface in $\mathbb{R}^3$ that minimizes the surface area to volume ratio.
Expressing $\Bbb N$ as an infinite union of disjoint infinite subsets.
Dual space $E'$ is metrizable iff $E$ has a countable basis
Is the ideal $I = \{f\mid f (0) = 0\}$ in the ring $C $ of all continuous real valued functions on $$ a maximal ideal?

I am trying to understand the definition of amalgamation of a $n+1$-partite graph as explained here(first few lines of page 4). We have a $n+1$-partite graph $G$ with partite sets $V_0,V_1,\cdots,V_n$ and we define amalgamation of $G$ on $V_i$ as follows: Let $W_i$ be a fixed set with $|W_i|=n|V_i|$. For each set, $A\subset W_i$ with $|A|=|V_i|$, we let $G_A$ be a “copy of G” with $i$th vertex class $V_i(G_A)=A$, all other *classes disjoint*. What does this mean? Classes will be disjoint for they are from different partite sets anyway.

I am trying to construct a small example as follows: Let $G$ be $K_{2,3,2}$. Suppose $V_0=\{v_1,v_2\}$, $V_1=\{w_1,w_2,w_3\}$ and $V_2=\{x_1,x_2\}$. Let $W_0=\{a_1,a_2,a_3,a_4\}$ and $A=\{a_1,a_2\}$. Now is $G_A$ the complete 3-partite graph with classes $\{a_1,a_2\},\{w_1,w_2,w_3\},\{x_1,x_2\}$?

The definition of amalgamation is as the union of all such graphs $G_A$. Does it mean the collection of all such $3$-partite graphs (visually speaking the graph obtained by drawing all such $3$-partite graphs side by side)?

- Alternating sum of a simple product of binomial coefficients: $\sum_{k=0}^{m} (-1)^k \binom m k \binom n k .$
- Restricted Compositions
- Ways to have exactly $n$ nominees out of $2n$ voters
- finding the combinatorial sum
- Multiple choice exam where no two consecutive answers are the same (2)
- Simplifying $\sum_{k=0}^{\lfloor{\frac{n}{2}}\rfloor}\binom{n}{2k}2^{2k}$

Thanks for your time.

- What do we call well-founded posets whose elements have a unique height?
- Show that if G is a simple graph with at least 4 vertices and 2n-3 edges, it must have two cycles of the same length.
- Finding combinatorial sum
- Is there a combinatorial way to see the link between the beta and gamma functions?
- Decomposition of graph to cycle and cut space
- How is $\dfrac1{(1-x)^5}=\sum_{n\geq0}{n+4\choose4}x^n$
- Find the expected number of two consecutive 1's in a random binary string
- Proof that $a^b>b^a$ if $a<b$ are integers larger or equal to two and $(a,b)\neq (2,3),(2,4)$
- A variant of the Knight's tour problem
- Is this graph connected

Leader’s description of $A_i(G)$ isn’t the clearest thing that I’ve ever read, but I think that I have it sorted.

Start with your $(n+1)$-partite graph $G$ with vertex classes $V_0,V_1,\dots,V_n$, and let $k$ be any positive integer. Fix an index $i$ with $0\le i\le n$; we’ll form the $k$-amalgamation of $G$ on $V_i$.

Let $W_i$ be a set of $k|V_i|$ new vertices. For each $A\subseteq W_i$ with $|A|=|V_i|$ form a graph $G_A$ as follows. Let $J=\{0,1,\dots,n\}\setminus\{i\}$. For each $j\in J$ let $V_j^A$ be a set of $|V_j|$ new vertices; the vertex set of $G_A$ is

$$A\cup\bigcup_{j\in J}V_j^A\;.$$

We fill in edges to make $G_A$ an isomorphic copy of $G$: the vertex classes of $G_A$ are $A$ and the sets $V_j^A$ for $j\in J$, and the isomorphism from $G_A$ to $G$ takes $A$ to $V_i$ and $V_j^A$ to $V_j$ for $j\in J$.

Now $W_i$ has $$\binom{k|V_i|}{|V_i}$$ subsets $A$ of cardinality $|V_i|$; call this number $m$. At this point we have $m$ copies of $G$, one for each $A\subseteq W_i$ of cardinality $|V_i|$. The $mn$ vertex classes in the family $$\left\{V_j^A:A\subseteq W_i\text{ and }|A|=|V_i|\text{ and }j\in J\right\}$$ are pairwise disjoint and are also disjoint from $W_i$.

Finally, $A_i(G)$ is the union of the $m$ graphs $G_A$: its vertex set is $$W_i\cup\bigcup\left\{V_j^A:A\subseteq W_i\text{ and }|A|=|V_i|\text{ and }j\in J\right\}\;,$$ so it has $$k|V_i|+m\sum_{j\in J}|V_j|$$ vertices. If $G$ has $e$ edges, each $G_A$ also has $e$ edges, and $A_i(G)$ therefore has $em$ edges.

If you start with $G=K_{2,3,2}$ and form the $2$-amalgamation on $V_0=\{v_1,v_2\}$, $W_0$ will be a $4$-element set, so you’ll have $m=\binom42=6$ copies $G_A$ of $G$, and $A_0(G)$, the amalgamation of these copies, will have $2|V_0|+mn=4+6\cdot(3+2)=34$ vertices.

- Every affine variety in $\mathbb A^n$ consisting of finitely many points can be written as the zero locus of $n$ polynomials
- Solution of Cauchy functional equation which has an antiderivative
- Intuition of Addition Formula for Sine and Cosine
- Show $\langle a^m \rangle \cap \langle a^n \rangle = \langle a^{\operatorname{lcm}(m, n)}\rangle$
- Arithmetic rules for big O notation, little o notation and so on…
- what is genus of complete intersection for: $F_1 = x_0 x_3 – x_1 x_2 , F_2 = x_0^2 + x_1^2 + x_2^2 + x_3^2$
- How do “Dummy Variables” work?
- Binomial probability with summation
- Priority vector and eigenvectors – AHP method
- Prove that any family of disjoint 8-signs on the plane is countable
- Is the configuration space of a connected space connected?
- Integral $\int\!\sqrt{\cot x}\,dx $
- Do $ AB $ and $ BA $ have same minimal and characteristic polynomials?
- Inequality constraints in calculus of variations
- Is a Fourier transform a change of basis, or is it a linear transformation?