2025/11/10 by Sudip Bera, Bera, Sudip
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Advanced Differential Equations and Dynamical Systems #Combinatorics (math.CO) #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2511.06689
openalex publication_date 2025/11/10 · openalex created_date 2025/11/12 · openalex updated_date 2026/07/28
The deep interconnection between linear algebra and graph theory allows one to interpret classical matrix invariants through combinatorial structures. To each square matrix A over a commutative ring K, one can associate a weighted directed graph D(A), where the algebraic behavior of A is reflected in the combinatorial properties of D(A). In particular, the determinant and characteristic polynomial of A admit elegant formulations in terms of sign-weighted sums over linear subdigraphs of D(A), thereby providing a graphical interpretation of fundamental algebraic quantities. Building upon this correspondence, we establish a combinatorial proof of the trace Cayley-Hamilton theorem. This theorem furnishes explicit trace identities linking the coefficients of the characteristic polynomial of A with the traces of its successive powers.