On minimal norms on MnM_n

In this note, we show that for each minimal norm $N(\cdot)$ on the algebra $M_n$ of all $n \times n$ complex matrices, there exist norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on ${\mathbb C}^n$ such that $$N(A)=\max\{\|Ax\|_2: \|x\|_1=1, x\in {\mathbb C}^n\}$$ for all $A \in M_n$. This may be regarded as an extension of a known result on characterization of minimal algebra norms.Comment: 4 pages, to appear in Abstract and Applied Analysi

A Characterization For 2-Self-Centered Graphs

A Graph is called 2-self-centered if its diameter and radius both equal to 2. In this paper, we begin characterizing these graphs by characterizing edge-maximal 2-self-centered graphs via their complements. Then we split characterizing edge-minimal 2-self-centered graphs into two cases. First, we characterize edge-minimal 2-self-centered graphs without triangles by introducing \emph{specialized bi-independent covering (SBIC)} and a structure named \emph{generalized complete bipartite graph (GCBG)}. Then, we complete characterization by characterizing edge-minimal 2-self-centered graphs with some triangles. Hence, the main characterization is done since a graph is 2-self-centered if and only if it is a spanning subgraph of some edge-maximal 2-self-centered graphs and, at the same time, it is a spanning supergraph of some edge-minimal 2-self-centered graphs

Function valued metric spaces

In this paper we introduce the notion of an ℱ-metric, as a function valued distance mapping, on a set X and we investigate the theory of ℱ-metrics paces. We show that every metric space may be viewed as an F-metric space and every ℱ-metric space (X,δ) can be regarded as a topological space (X,τδ). In addition, we prove that the category of the so-called extended F-metric spaces properly contains the category of metric spaces. We also introduce the concept of an `ℱ-metric space as a completion of an ℱ-metric space and, as an application to topology, we prove that each normal topological space is `ℱ-metrizable