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Delio Jaramillo Velez (Virginia Tech)5/18/26, 11:00 AMAlgebraic Invariants of GraphsMinisymposium Talk
A connected dominating set of a graph is a vertex set that induces a connected subgraph and such that every vertex outside the set is adjacent to at least one vertex in the set. The minimum cardinality of a connected dominating set is called the connected domination number. We present an algebraic expression for this combinatorial invariant using the theory of binomial edge ideals.
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Juan Pablo Serrano Perez (Cinvestav)5/18/26, 11:25 AMAlgebraic Invariants of GraphsMinisymposium Talk
The distance ideals of connected graphs are algebraic invariants extending the Smith normal form (SNF) and the spectrum of graph distance matrices.
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In general, distance ideals are not monotone under taking induced subgraphs.
However, it was proved in 2017 that the set of graphs with one trivial distance ideal over $\mathbb{Z}[X]$ and over $\mathbb{Q}[X]$ was characterized in terms... -
Antwon Park (University of Kentucky)5/18/26, 11:50 AMAlgebraic Invariants of GraphsMinisymposium Talk
We introduce the family of graphical Hermite simplices and study the Smith normal forms of their matrices of vertex vectors, which is equivalent to studying the group structure of the cokernels for these matrices.
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Our motivation is to study the behavior of lattice simplices subject to small lattice perturbations of their vertices.
In this case, a graphical Hermite simplex is a perturbation... -
Joel Louwsma (Niagara University)5/18/26, 2:00 PMAlgebraic Invariants of GraphsMinisymposium Talk
Chip firing provides a way to study the sandpile group (also known as the Jacobian) of a graph. We use a generalized version of chip firing to bound the number of invariant factors of the critical group of an arithmetical structure on a graph. We also show that, under suitable hypotheses, critical groups are additive under wedge sums of graphs with arithmetical structures. These results allow...
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Dr Carlos Alfaro (Banxico)5/18/26, 2:25 PMAlgebraic Invariants of GraphsMinisymposium Talk
Graham, Lovász and Pollak obtained a well known formula for the determinant of distance matrices of trees. This formula depends only on the number of vertices of the tree and not on its topological structure. Later, Hou and Woo computed the explicit expression of the Smith form of the distance matrix of a tree, which again, it depends only on the number of vertices. In this talk, we will show...
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Ralihe Raul Villagran Olivas (Worcester Polytechnic Institute)5/18/26, 2:50 PMAlgebraic Invariants of GraphsMinisymposium Talk
Chip-firing is a discrete dynamical process on a graph that exhibits striking phenomena, including fractal-like symmetries and self-organized criticality. From an algebraic perspective, the states of this process (combinatorially) define a group, the Sandpile group, whose algebraic structure is given by the Smith normal form of the Laplacian matrix. We will discuss problems related to Sandpile...
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Kate Lorenzen (Linfield University)5/19/26, 11:00 AMAlgebraic Invariants of GraphsMinisymposium Talk
Graphs can be encoded into a matrix according to some rule. The eigenvalues of the matrix are used to understand the structural properties of graphs. If two graphs share a set of eigenvalues, they are called cospectral. A tree is a graph with no cycles, and for most matrix representations, almost all trees have a cospectral mate. The distance Laplacian matrix is found by subtracting the...
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Prof. Vilmar Trevisan (UFRGS - Brazil)5/19/26, 11:25 AMAlgebraic Invariants of GraphsMinisymposium Talk
Within Spectral Graph Theory, Brouwer’s Conjecture (BC) is a fundamental problem concerning Laplacian eigenvalues and graph invariants. It proposes a relationship between the sum of the largest Laplacian eigenvalues of a graph and its number of edges, with direct implications for the study of Laplacian energy. More precisely, for a graph ( G = (V, E) ) with ( n = |V| ) vertices and ( m = |E| )...
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Colby Sherwood (University of Delaware)5/19/26, 11:50 AMAlgebraic Invariants of GraphsMinisymposium Talk
Let $W^i_{k,n}(m)$ denote a matrix with rows and columns indexed
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by the $k$-subsets and $n$-subsets, respectively, of an $m$-element set. The row $S$, column $T$ entry of $W^i_{k,n}(m)$ is 1 if $|S \cap T|= i$, and is 0 otherwise. When $i=k$ the matrix $W^k_{k,n}(m)$ is the subset inclusion matrix for which Wilson found a diagonal form, solving the $p$-rank problem for any prime $p$. This...
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