There are many ways to associate a graph (combinatorial structure) to a
commutative ring $R$ with unity. One of the ways is to associate a
*zero-divisor graph* $\Gamma(R)$ to $R$. The vertices of $\Gamma(R)$
are all elements of $R$ and two vertices $x, y \in R$ are adjacent in
$\Gamma(R)$ if and only if $xy = 0$. We shall discuss Laplacian of a
combinatorial structure $\Gamma(R)$ and show that the representatives of
some algebraic invariants are eigenvalues of the Laplacian of
$\Gamma(R)$. Moreover, we discuss association of another combinatorial
structure (Young diagram) with $R$. Let $m, n \in \mathbb{Z}_{>0}$ be two
positive integers. The Young’s partition lattice $L(m,n)$ is defined to
be the poset of integer partitions $\mu = (0 \leq \mu_1 \leq \mu_2 \leq
\cdots \leq \mu_m \leq n)$. We can visualize the elements of this poset
as Young diagrams ordered by inclusion. We conclude this talk with a
discussion on Stanley’s conjecture regarding symmetric saturated chain
decompositions (SSCD) of $L(m,n)$.

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Last updated: 19 Jul 2024