Let \(G\) be a simple graph with vertex set \(V=\{v_1,v_2,\ldots,v_n \}\), and let \(d_i\) be the degree of the vertex \(v_i\). The Sombor matrix of \(G\) is the square matrix \(\mathbf A_{SO}\) of order \(n\), whose \((i,j)\)-element is \(\sqrt{d_i^2+d_j^2}\) if \(v_i\) and \(v_j\) are adjacent, and zero otherwise. We study the characteristic polynomial, spectrum, and energy of \(\mathbf A_{SO}\). A few results for the coefficients of the characteristic polynomial, and bounds for the energy of \(\mathbf A_{SO}\) are established.

This note presents the characterization of the families of star, helm, flower and complete graphs by total vertex stress. The note does not present results for many families of graphs but, it highlights important philosophical (math. phil.) aspects for further research. In particular the novelty concepts of forgiven contradictions denoted by, iff\(_f\) as well as iffness and \(f\)-statements are introduced. The author suggests that the characterization of other families of graphs by total vertex stress is possible.

In this paper, we define some new operators \([(A \$ B),(A \# B),(A\ast B),(A \rightarrow B) ]\) of Fermatean fuzzy matrices and investigate their algebraic properties. Further, the necessity and possibility operators of Fermatean fuzzy matrices are proved. Finally, we have identified and proved several of these properties, particularly those involving the operator \(A\rightarrow B\) defined as Fermatean fuzzy implication with other operators.

In this paper, by using \(S\)-norms, we defined anti fuzzy subgroups and anti fuzzy normal subgroups which are new notions and considered their fundamental properties and also made an attempt to study the characterizations of them. Next we investigated image and pre image of them under group homomorphisms. Finally, we introduced the direct sum of them and proved that direct sum of any family of them is also anti fuzzy subgroups and anti fuzzy normal subgroups under \(S\)-norms, respectively.