On the Spectra of Simplicial Rook Graphs

On Isomorphism of Simplicial Complexes and Their Related Algebras

Vertex Decomposable Simplicial Complexes Associated to Path Graphs

Introduction Vertex decomposability of a simplicial complex is a combinatorial topological concept which is related to the algebraic properties of the Stanley-Reisner ring of the simplicial complex. This notion was first defined by Provan and Billera in 1980 for k-decomposable pure complexes which is known as vertex decomposable when . Later Bjorner and Wachs extended this concept to non-pure ...

Notes on simplicial rook graphs

The simplicial rook graph SR(m,n) is the graph of which the vertices are the sequences of nonnegative integers of length m summing to n, where two such sequences are adjacent when they differ in precisely two places. We show that SR(m,n) has integral eigenvalues, and smallest eigenvalue s = max(−n,− ( m 2 ) ), and that this graph has a large part of its spectrum in common with the Johnson graph...

SPECTRAL AND COMBINATORIAL PROPERTIES OF FRIENDSHIP GRAPHS, SIMPLICIAL ROOK GRAPHS, AND EXTREMAL EXPANDERS by

Algebraic combinatorics is the area of mathematics that uses the theories and methods of abstract and linear algebra to solve combinatorial problems, or conversely applies combinatorial techniques to solve problems in algebra. In particular, spectral graph theory applies the techniques of linear algebra to study graph theory. Spectral graph theory is the study of the eigenvalues of various matr...

Shifted Simplicial Complexes Are Laplacian Integral

We show that the combinatorial Laplace operators associated to the boundary maps in a shifted simplicial complex have all integer spectra. We give a simple combinatorial interpretation for the spectra in terms of vertex degree sequences, generalizing a theorem of Merris for graphs. We also conjecture a majorization inequality for the spectra of these Laplace operators in an arbitrary simplicial...