in this paper some upper and lower bounds for the greatest eigenvalues of the pi and vertex pimatrices of a graph g are obtained. those graphs for which these bounds are best possible arecharacterized.

let ‎x be an ‎‎n-‎‎‎‎‎‎square complex matrix with the ‎cartesian decomposition ‎‎x = a + i ‎b‎‎‎‎‎, ‎where ‎‎a ‎and ‎‎b ‎are ‎‎‎n ‎‎times n‎ ‎hermitian ‎matrices. ‎it ‎is ‎known ‎that ‎‎$vert x vert_p^2 ‎leq 2(vert a vert_p^2 + vert b vert_p^2)‎‎‎$, ‎where ‎‎$‎p ‎‎geq 2‎$‎ ‎and ‎‎$‎‎vert . vert_p$ ‎is ‎the ‎schatten ‎‎‎‎p-norm.‎ ‎‎ ‎‎in this paper‎, this inequality and some of its improvements ...

In this paper, we discuss some properties of joint spectral {radius(jsr)} and  generalized spectral radius(gsr)  for a finite set of upper triangular matrices with entries in a Banach algebra and represent relation between geometric and joint/generalized spectral radius. Some of these are in scalar matrices, but  some are different. For example for a bounded set of scalar matrices,$Sigma$, $r_*...

The lower spectral radius, or joint spectral subradius, of a set of real d× d matrices is defined to be the smallest possible exponential growth rate of long products of matrices drawn from that set. The lower spectral radius arises naturally in connection with a number of topics including combinatorics on words, the stability of linear inclusions in control theory, and the study of random Cant...

The product of two real spectral triples {A1,H1,D1,J1, γ1} and {A2,H2,D2,J2(, γ2)}, the first of which is necessarily even, was defined by A.Connes [3] as {A,H,D,J (, γ)} given by A = A1 ⊗ A2, H = H1 ⊗ H2, D = D1 ⊗ Id2 + γ1 ⊗ D2, J = J1 ⊗ J2 and, in the even-even case, by γ = γ1⊗γ2. Generically it is assumed that the real structure J obeys the relations J 2 = ǫId, JD = ǫ ′DJ , J γ = ǫ ′′γJ , wh...

Two characterizations of general matrices for which the spectral radius is an eigenvalue and the corresponding eigenvector is either positive or nonnegative are presented. One is a full characterization in terms of the sign of the entries of the spectral projector. In another case, different necessary and sufficient conditions are presented that relate to the classes of the matrix. These charac...

In this paper some upper and lower bounds for the greatest eigenvalues of the PI and vertex PI matrices of a graph G are obtained. Those graphs for which these bounds are best possible are characterized.