Determinantal Ideals of Graphs

Carlos A. Alfaro (), Juan Pablo Serrano () and Ralihe R. Villagrán ()
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Carlos A. Alfaro: Banco de México
Juan Pablo Serrano: Centro de Investigación y de Estudios Avanzados del IPN, Departamento de Matemáticas
Ralihe R. Villagrán: Worcester Polytechnic Institute, Department of Mathematical Sciences

A chapter in Handbook of Visual, Experimental and Computational Mathematics, 2026, pp 813-844 from Springer

Abstract: Abstract Determinantal ideals of graphs are the ideals generated by the k-minors of matrices whose entries are in a polynomial ring obtained from the combinatorial properties of a graph. Determinantal ideals have been the central topic in both commutative algebra and algebraic geometry, and they also have numerous connections with invariant theory, representation theory, and combinatorics. These determinantal ideals generalize the spectrum and the Smith normal form (SNF) of matrices associated with graphs. Since matrices associated with graphs play a crucial role in designing networks, developing routing algorithms, and optimizing data transmission, then the reader will find in determinantal ideals promising applications in network theory, optimization, complex networks, among others. This brief survey focuses on two determinantal ideals: critical ideals and distance ideals. Critical ideals have been used to obtain characterizations of graphs with a fixed number of generators and to calculate the sandpile groups of some families of graphs. Furthermore, there are relations between critical ideals and the concepts of zero-forcing number and minimum rank. On the other hand, in this survey, characterizations of graphs and digraphs with one trivial distance ideal are given. These characterizations have interesting relations with well-known families of graphs, like distance-hereditary and perfect graphs. Also, the concept of cospectrality can be extended to determinantal ideals.

Keywords: Determinantal ideals; Critical ideals; Distance ideals; Graphs; Adjacency matrix; Laplacian matrix; Distance matrix (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1007/978-3-032-16368-4_75

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