Vol 30, No 150 (2025)

Abstract. In the article, the authors consider the problem of constructing an upper bound for the sum of the maximal eigenvalues of Laplacian of a graph. The article is devoted to proving the Brouwer conjecture, which states that the sum of the -maximal eigenvalues of Laplacian of a graph does not exceed the number of edges of the graph plus \( (t + 1)t⁄2 \). Note that we prove the validity of the general Brouwer conjecture under the assumption that the conjecture is valid for a finite number of graphs with the number of vertices less than \( 10^{24 } \), i.e., a complete proof of the conjecture is reduced to establishing its validity for a finite number of graphs. The proof of this conjecture attracts the interest of a large number of specialists. There are a number of results for special graphs and a proof of the conjecture for almost all random graphs. The proof we are considering uses an inductive method that has some peculiarities. The original method involves constructing various estimates for the eigenvalues of Laplacian of a graph which is used to construct the induction step. Several variants of the method are considered depending on the values of the coordinates of the eigenvectors of the Laplacian. The well-known fact of equivalence of the validity of the Brouwer conjecture for the graph itself and the complement of the graph is used.

The paper calculates the structure of the kernel and co-kernel of the Schwartz problem for $J$-analytic functions defined in the ellipse $D$ with a boundary $\Gamma.$ The Schwartz problem consists in finding a $J$-analytic function in the ellipse $D$ by the known value of its real part on $\Gamma. $ In paragraphs 1 and 2 the problem is formulated and its solution for a~special right part is studied. Paragraph 3 contains the necessary information from one paper by A.\,P.~Soldatov. Paragraph 4 constructs the solution of the Schwarz union problem for the special right-hand side. On the basis of these results, paragraph 5 calculates the kernel and the co-kernel of the Schwartz problem. The model of their calculation is briefly described at the beginning of the fifth paragraph. Then in the theorems \ref{th5.1}--\ref{th5.6} this scheme is implemented. Here the notions of theoretical and algorithmic solvability of the special Schwarz problem introduced by the author are used. The method of mathematical induction is used as well. It is shown that the kernel and co-kernel of the Schwarz problem in an ellipse consist only of vector polynomials. The paper describes the structure of the kernel and co-kernel in terms of the ranks of some real matrices depending on the matrix $J$ and the ellipse $\Gamma.$ The paper concludes with an example of calculating the kernel of the Schwarz problem in an ellipse for a two-dimensional matrix $J$ with multiple eigenvalue.

or non-autonomous differential inclusions, the issues of attraction and asymptotic behavior of solutions are considered. The basis of the research is the development of the method of limit differential equations in combination with the direct Lyapunov method with several Lyapunov functions. This makes it possible to more accurately localize and determine the structure of \( ω \)-limit sets of solutions. The main problems of the research are the absence of properties of the invariance type of \( ω \)-limit sets of non-autonomous systems and the construction of limit differential relations. They are solved using limit differential inclusions constructed using shifts (translations) of the main differential inclusions. The results have the form of generalizations of the LaSalle invariance principle and provide preliminary information on the limit behavior of solutions. A set of additional Lyapunov functions allows one to refine this behavior and to single out those points from the set of zeros of the derivative of the main Lyapunov function that obviously do not belong to the \( ω \)-limit sets. The results are illustrated by the example of a linear oscillator with dry friction.