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The group of commutativity preserving maps on strictly upper triangular matrices

Deng Yin Wang, Min Zhu, Jianling Rou (2014)

Czechoslovak Mathematical Journal

Let 𝒩 = N n ( R ) be the algebra of all n × n strictly upper triangular matrices over a unital commutative ring R . A map ϕ on 𝒩 is called preserving commutativity in both directions if x y = y x ⇔ ϕ ( x ) ϕ ( y ) = ϕ ( y ) ϕ ( x ) . In this paper, we prove that each invertible linear map on 𝒩 preserving commutativity in both directions is exactly a quasi-automorphism of 𝒩 , and a quasi-automorphism of 𝒩 can be decomposed into the product of several standard maps, which extains the main result of Y. Cao, Z. Chen and C. Huang (2002) from fields to rings.

The higher rank numerical range of nonnegative matrices

Aikaterini Aretaki, Ioannis Maroulas (2013)

Open Mathematics

In this article the rank-k numerical range ∧k (A) of an entrywise nonnegative matrix A is investigated. Extending the notion of elements of maximum modulus in ∧k (A), we examine their location on the complex plane. Further, an application of this theory to ∧k (L(λ)) of a Perron polynomial L(λ) is elaborated via its companion matrix C L.

The inertia set of nonnegative symmetric sign pattern with zero diagonal

Yubin Gao, Yan Ling Shao (2003)

Czechoslovak Mathematical Journal

The inertia set of a symmetric sign pattern A is the set i ( A ) = { i ( B ) ∣ B = B T ∈ Q ( A ) } , where i ( B ) denotes the inertia of real symmetric matrix B , and Q ( A ) denotes the sign pattern class of A . In this paper, a complete characterization on the inertia set of the nonnegative symmetric sign pattern A in which each diagonal entry is zero and all off-diagonal entries are positive is obtained. Further, we also consider the bound for the numbers of nonzero entries in the nonnegative symmetric sign patterns A with zero diagonal that require...

The k-Fibonacci matrix and the Pascal matrix

Sergio Falcon (2011)

Open Mathematics

We define the k-Fibonacci matrix as an extension of the classical Fibonacci matrix and relationed with the k-Fibonacci numbers. Then we give two factorizations of the Pascal matrix involving the k-Fibonacci matrix and two new matrices, L and R. As a consequence we find some combinatorial formulas involving the k-Fibonacci numbers.

The Laplacian spectrum of some digraphs obtained from the wheel

Li Su, Hong-Hai Li, Liu-Rong Zheng (2012)

Discussiones Mathematicae Graph Theory

The problem of distinguishing, in terms of graph topology, digraphs with real and partially non-real Laplacian spectra is important for applications. Motivated by the question posed in [R. Agaev, P. Chebotarev, Which digraphs with rings structure are essentially cyclic?, Adv. in Appl. Math. 45 (2010), 232-251], in this paper we completely list the Laplacian eigenvalues of some digraphs obtained from the wheel digraph by deleting some arcs.

The Laplacian spread of graphs

Zhifu You, Bo Lian Liu (2012)

Czechoslovak Mathematical Journal

The Laplacian spread of a graph is defined as the difference between the largest and second smallest eigenvalues of the Laplacian matrix of the graph. In this paper, bounds are obtained for the Laplacian spread of graphs. By the Laplacian spread, several upper bounds of the Nordhaus-Gaddum type of Laplacian eigenvalues are improved. Some operations on Laplacian spread are presented. Connected c -cyclic graphs with n vertices and Laplacian spread n - 1 are discussed.

The Least Eigenvalue of Graphs whose Complements Are Uni- cyclic

Yi Wang, Yi-Zheng Fan, Xiao-Xin Li, Fei-Fei Zhang (2015)

Discussiones Mathematicae Graph Theory

A graph in a certain graph class is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum among all graphs in that class. Bell et al. have identified a subclass within the connected graphs of order n and size m in which minimizing graphs belong (the complements of such graphs are either disconnected or contain a clique of size n/2 ). In this paper we discuss the minimizing graphs of a special class of graphs of order n whose complements are connected and contains...

The Legendre Formula in Clifford Analysis

Laville, Guy, Ramadanoff, Ivan (2009)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 30A05, 33E05, 30G30, 30G35, 33E20.Let R0,2m+1 be the Clifford algebra of the antieuclidean 2m+1 dimensional space. The elliptic Cliffordian functions may be generated by the z2m+2 function, analogous to the well-known Weierstrass z-function. The latter satisfies a Legendre equality. We prove a corresponding formula at the level of the monogenic function Dm z2m+2.

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