On The Spread Of Real Symmetric Matrices With Entries In An Interval

A spread of a matrix has extensive and practical applications in the combinatorial optimization problems and cybernetics problems. Many bibliographies have researched the bounds of the spread of a matrix but none of them refers to the spread of real symmetric matrices, order n, with entries in the interval [a,b]. We are interested in this case. Firstly, we find the entries are nothing but a or b if the spread of the matrix is maximum. Next, when the spread attains to the upper bound of Mirsky's theorem, we describe the structure character of the matrices and give some examples. Then we focus our study on the maximal value of the spread and the corresponding structure of matrices whose entries are a or 1 and whose rank is 2. In the following, we bring up conjectures about the expression of spread and the corresponding structure of matrices whose entries are in the interval [a,b] if the spread acquires the maximal value. At last we verify the results by the Matlab experiments.