Commuting graphs of matrices over semirings

On idempotent matrices over semirings

Idempotent matrices play a significant role while dealing with different questions in matrix theory and its applications. It is easy to see that over a field any idempotent matrix is similar to a diagonal matrix with 0 and 1 on the main diagonal. Over a semiring the situation is quite different. For example, the matrix J of all ones is idempotent over Boolean semiring. The first characterizatio...

commuting and non -commuting graphs of finit groups

فرض کنیمg یک گروه غیر آبلی متناهی باشد . گراف جابجایی g که با نماد نمایش داده می شود ،گرافی است ساده با مجموعه رئوس که در آن دو راس با یک یال به هم وصل می شوند اگر و تنها اگر . مکمل گراف جابجایی g راگراف نا جابجایی g می نامیم.و با نماد نشان می دهیم. گرافهای جابجایی و ناجابجایی یک گروه متناهی ،اولین بار توسطاردوش1 مطرح گردید ،ولی در سالهای اخیر به طور مفصل در مورد بحث و بررسی قرار گرفتند . در ،م...

Multiplicative Equations over Commuting Matrices (extended Abstract)

We consider the solvability of the equation k Yi=1Aixi = B and generalizations, where the Ai and B are given commutingmatrices over an algebraic number eld F . In the semigroup membership problem, the variables xi are constrained to be nonnegative integers. While this problem is NP-complete for variable k, we give a polynomial time algorithm if k is xed. In the group membership problem, the mat...

Efficient transitive closure of sparse matrices over closed semirings

Tan's Epsilon-Determinant and Ranks of Matrices over Semirings

We use the ϵ-determinant introduced by Ya-Jia Tan to define a family of ranks of matrices over certain semirings. We show that these ranks generalize some known rank functions over semirings such as the determinantal rank. We also show that this family of ranks satisfies the rank-sum and Sylvester inequalities. We classify all bijective linear maps which preserve these ranks.