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Two graphs having the same spectrum are said to be cospectral. Two graphs such that the absolute values of their nonzero eigenvalues coincide are singularly cospectral graphs. Cospectrality implies singular cospectrality, but the converse may be false. In this paper, we present sufficient conditions for two circulant graphs, with an even number of vertices, to be noncospectral singularly cospectral graphs. In this analysis, we study when a pair of these graphs have the same or distinct inertia. In addition, we show that two singularly cospectral circulant graphs with an odd prime number of vertices are isomorphic.

A signed graph is one that features two types of edges: positive and negative. Balanced signed graphs are those in which all cycles contain an even number of positive edges. In the adjacency matrix of a signed graph, entries can be $0$, $-1$, or $1$, depending on whether $ij$ represents no edge, a negative edge, or a positive edge, respectively. The index of the adjacency matrix of a signed graph $\dot{G}$ is less or equal to the index of the adjacency matrix of its underlying graph $G$, i.e., $\lambda_1(\dot{G}) \le \lambda_1(G)$. Indeed, if $\dot{G}$ is balanced, then $\lambda_1(\dot{G})=\lambda_1(G)$. This inequality becomes strict when $\dot{G}$ is an unbalanced signed graph. Recently, Brunetti and Stani\'c found the whole list of unbalanced signed graphs on $n$ vertices with maximum (resp. minimum) spectral radius. To our knowledge, there has been little research on this problem when unbalanced signed graphs are confined to specific graph classes. In this article, we demonstrate that there is only one unbalanced signed bipartite graph on $n$ vertices with maximum spectral radius, up to an operation on the signed edges known as switching. Additionally, we investigate unbalanced signed complete bipartite graphs on $n$ vertices with a bounded number of edges and maximum spectral radius, where the negative edges induce a tree.

The main study of this article is the characterization of Richard's inequality, because it is closely related to Buzano's inequality. Finally, we present a newapproach for Richard's inequality, where we use the Selberg operator.

M. Lin defined a binary operation for two positive semi-definite matrices in studying certain determinantal inequalities that arise from diffusion tensor imaging. This operation enjoys some interesting properties similar to the operator geometric mean. We study this operation further and present numerous properties emphasizing the relationship with the operator geometric mean. In the end, we present an application toward Tsallis relative operator entropy., Comment: 17 pages

This paper explores the concept of approximate Birkhoff-James orthogonality in the context of operators on semi-Hilbert spaces. These spaces are generated by positive semi-definite sesquilinear forms. We delve into the fundamental properties of this concept and provide several characterizations of it. Using innovative arguments, we extend a widely known result initially proposed by Magajna in [J. London. Math. Soc., 1993]. Additionally, we improve a recent result by Sen and Paul in [Math. Slovaca, 2023] regarding a characterization of approximate numerical radius orthogonality of two semi-Hilbert space operators, such that one of them is $A$-positive. Here, $A$ is assumed to be a positive semi-definite operator., Comment: 17 pages

This article implements a simple convex approach and block techniques to obtain several new refined versions of numerical radius inequalities for Hilbert space operators. This includes comparisons among the norms of the operators, their Cartesian parts, their numerical radii, the numerical radius of the product of two operators, and the Aluthge transform.

In this paper, we introduce the $f-$operator radius of Hilbert space operators as a generalization of the Euclidean operator radius and the $q-$operator radius. Properties of the newly defined radius are discussed, emphasizing how it extends some known results in the literature.

The main goal of this work is to present new matrix inequalities of the Cauchy-Schwarz type. In particular, we investigate the so-called Lieb functions, whose definition came as an umbrella of Cauchy-Schwarz-like inequalities, then we consider the mixed Cauchy-Schwarz inequality. This latter inequality has been influential in obtaining several other matrix inequalities, including numerical radius and norm results. Among many other results, we show that \[\left\| T \right\|\le \frac{1}{4}\left( \left\| \left| T \right|+\left| {{T}^{*}} \right|+2\mathfrak RT \right\|+\left\| \left| T \right|+\left| {{T}^{*}} \right|-2\mathfrak RT \right\| \right),\] where $\mathfrak RT$ is the real part of $T$., Comment: to appear in Oper. Matrices

If $P$ is an orthogonal projection defined on an inner product space $\mathcal{H}$, then the inequality $$ |\langle Px, y\rangle|\leq \frac12 [\|x\|\|y\|+|\langle x, y\rangle|] $$ fulfills for any $x,y \in \mathcal{H}$ (see \cite{Dra16}). In particular, when $P$ is the identity operator, then it recovers the famous Buzano inequality. We obtain generalizations of such classical inequality, which hold for certain families of bounded linear operators defined on $\mathcal{H}$. In addition, several new inequalities involving the norm and numerical radius of an operator are established., Comment: 15 pages

In this article, we present some new inequalities for the numerical radius of products of Hilbert space operators and the generalized Aluthge transform. In particular, we show some upper bounds for $\omega(ABC+DEF)$ using the celebrated Buzano inequality, then some consequences that generalize some results from the literature are discussed. After that, inequalities that involve the generalized Aluthge transform are shown using some known bounds for the numerical radius of the product of two operators.

The weighted numerical radius of a Hilbert space operator has been defined recently. This article explores other properties and uses this newly defined numerical radius to obtain several new interesting inequalities for the weighted numerical radius and the numerical radius. In particular, new identities for the numerical radius, further comparisons between the numerical radius and the real and imaginary parts, and some inequalities for sectorial operators will be presented.