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This paper lies in the context of the studies of eigenfunctions of graphs having minimum cardinality of support. One of the tools is the weight‐distribution bound, a lower bound on the cardinality of support of an eigenfunction of a distance‐regular graph corresponding to a nonprincipal eigenvalue. The tightness of the weight‐distribution bound was previously shown in general for the smallest eigenvalue of a Grassmann graph. However, a characterisation of optimal eigenfunctions was not obtained. Motivated by this open problem, we consider the class of strongly regular Grassmann graphs and give the required characterisation in this case. We then show the tightness of the weight‐distribution bound for block graphs of affine designs (defined on the lines of an affine space with two lines being adjacent when intersect) and obtain a similar characterisation of optimal eigenfunctions. [ABSTRACT FROM AUTHOR]

Introduction. The relevance of the study is linked to the unsustainable trend of increasing waste volumes on a global scale. Ensuring sustainable development of any state is accompanied by the creation of a safe environment for present and future generations. Since the state is called upon to guarantee safe living conditions for its citizens, it is advisable to analyze the experience of foreign countries in implementing management measures in the sphere of solid waste. The research is based mainly on the publications of foreign authors and organizations, as well as statistical data of foreign information publications. Purpose. Since solid waste collection and disposal is one of the major problems in most of the world today, the purpose of the article is to study the experience of foreign countries in the field of public management of solid waste. Results. In the article the world tendencies of the formation and accumulation of wastes are analyzed. The analysis covers countries of such regions as East Asia and the Pacific, Europe, Central and South Asia, Latin and North America, the Caribbean, the Middle East, North Africa and Africa, south of Sahara. It is estimated that the largest volumes of waste are generated in East Asia and the Pacific – 23%, and the least in the Middle East and North Africa – 6%. The main methods and systems of public waste management in foreign countries are analyzed in the context of their levels of development and their provision with relevant regulations. Conclusions. The most progressive countries in the world on issues of public solid waste management are: Sweden, Switzerland, Germany, France, Poland, Austria, Great Britain, Finland, USA and Japan. The features of solid waste management in these countries are described. These countries have established a system of regulations as well as a network of specialized environmental agencies that provide a wide range of services in the field of solid waste management. Over the decades, these countries have gained experience in waste management by improving their waste management practices according to their specific requirements. In other countries, there are significant gaps in legislation, environmental awareness and the organization of solid waste management activities.

A k‐regular graph is called a divisible design graph (DDG) if its vertex set can be partitioned into m classes of size n, such that two distinct vertices from the same class have exactly λ1 common neighbours, and two vertices from different classes have exactly λ2 common neighbours. A DDG with m=1, n=1, or λ1=λ2 is called improper, otherwise it is called proper. We present new constructions of DDGs and, using a computer enumeration algorithm, we find all proper connected DDGs with at most 39 vertices, except for three tuples of parameters: (32,15,6,7,4,8), (32,17,8,9,4,8), and (36,24,15,16,4,9). [ABSTRACT FROM AUTHOR]