in this paper, we present some inequalities for the co-pi index involving the some topological indices, the number of vertices and edges, and the maximum degree. after that, we give a result for trees. in addition, we give some inequalities for the largest eigenvalue of the co-pi matrix of g.

We systematically develop real Paley–Wiener theory for the Fourier transform on Rd for Schwartz functions, Lp-functions and distributions, in an elementary treatment based on the inversion theorem. As an application, we show how versions of classical Paley–Wiener theorems can be derived from the real ones via an approach which does not involve domain shifting and which may be put to good use fo...

The purpose of this paper is to present a unified Perron-Frobenius Theory for nonnegative, for real not necessarily nonnegative and for general complex matrices. The sign-real spectral radius was introduced for general real matrices. This quantity was shown to share certain properties with the Perron root of nonnegative matrices. In this paper we introduce the sign-complex spectral radius. Agai...

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 ...

We study the finite-step realizability of the joint/generalized spectral radius of a pair of real d × d matrices {S1, S2}, one of which has rank 1, where 2 ≤ d < +∞. Let ρ(A) denote the spectral radius of a square matrix A. Then we prove that there always exists a finite-length word (i1, . . . , i ∗ l) ∈ {1, 2}, for some finite l ≥ 1, such that l √ ρ(Si1 · · ·Si∗l ) = sup n≥1 { max (i1,...,in)∈...