The main vertices of a star set and related graph parameters

A vertex v is an element of V (G) is called lambda-main if it belongs to a star set X subset of V (G) of the eigenvalue lambda of a graph G and this eigenvalue is main for the graph obtained from G by deleting all the vertices in X \ {v}; otherwise, v is lambda-non-main. Some results concerning main and non-main vertices of an eigenvalue are deduced. For a main eigenvalue lambda of a graph G, we introduce the minimum and maximum number of lambda-main vertices in some lambda-star set of G as new graph invariant parameters. The determination of these parameters is formulated as a combinatorial optimization problem based on a simplex-like approach. Using these and some related parameters we develop new spectral tools that can be used in the research of the isomorphism problem. Examples of graphs for which the maximum number of lambda-main vertices coincides with the cardinality of a lambda-star set are provided. (C) 2021 Elsevier B.V. All rights reserved.