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.

Abstract. For a connected graph, the distance spectral radius is the largest eigenvalue of its distance matrix. Let U1 2k be the graph obtained from C3 by attaching a path of length n− 3 at one vertex. Let U2 2k be the graph obtained from C3 by attaching a pendant edge together with k − 2 paths of length 2 at the same vertex. In this paper, it is proved that U1 2k (resp., U 2 2k) is the unique ...

It is shown that a transitive, closed, homogeneous semigroup of linear transformations on a finite-dimensional space either has zero divisors or is simultaneously similar to a group consisting of scalar multiples of unitary transformations. The proof begins with the result that for each closed homogeneous semigroup with no zero divisors there is a k such that the spectral radius satisfies r(AB)...

Abstract It is known that the spectral radius of a digraph with k edges is ≤ √ k, and that this inequality is strict except when k is a perfect square. For k = m + l, l fixed, m large, Friedland showed that the optimal digraph is obtained from the complete digraph on m vertices by adding one extra vertex and connecting it to the first l/2 vertices by pairs of directed edges. Using a combinatori...

Examples: 1) Local spectral radius r(T, x) of an operator T at a point x ∈ X can be defined by r(T, x) = lim supk→∞ ‖T kx‖1/k, i.e. it is a quantity defined in terms of B. The local spectral radius plays an important role in the local spectral theory. 2) As an analogy to the local spectral radius for the set of all polynomials can be considered the local capacity (see later). 3) The invariant s...

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 classify the growth of a k-regular sequence based on information from its k-kernel. In order to provide such a classification, we introduce the notion of a growth exponent for k-regular sequences and show that this exponent is equal to the joint spectral radius of any set of a special class of matrices determined by the k-kernel.