An eigenvalue of a real symmetric matrix is called main if there is an associated eigenvector not orthogonal to the all-1 vector . Main eigenvalues are frequently considered in the framework of simple undirected graphs. In this study we generalize some results and then apply them to signed graphs.
We show positivity of the Q-matrix of four kinds of graph products: direct product (Cartesian product), star product, comb product, and free product. During the discussion we give an alternative simple proof of the Markov product theorem on positive definite kernels.
We will discuss how graph based matrices are capable to find classification of the graph vertices with small within- and between-cluster discrepancies. The structural eigenvalues together with the corresponding spectral subspaces of the normalized modularity matrix are used to find a block-structure in the graph. The notions are extended to rectangular arrays of nonnegative entries and to directed graphs. We also investigate relations between spectral properties, multiway discrepancies, and degree...
Zero forcing number has recently become an interesting graph parameter studied in its own right since its introduction by the “AIM Minimum Rank–Special Graphs Work Group”, whereas metric dimension is a well-known graph parameter. We investigate the metric dimension and the zero forcing number of some line graphs by first determining the metric dimension and the zero forcing number of the line graphs of wheel graphs and the bouquet of circles. We prove that for a simple and connected graph . Further,...
Metrically regular bigraphs the square of which are metrically regular graphs are investigated in the case of graphs with 6 distinct eigenvalues (these eigenvalues can have variuos multiplicities).
The present paper deals with the spectra of powers of metrically regular graphs. We prove that there is only two tables of the parameters of an association scheme so that the corresponding metrically regular bipartite graph of diameter (8 distinct eigenvalues of the adjacency matrix) has the metrically regular square. The results deal with the graphs of the diameter see [8], [9] and [10].
The present paper deals with the spectra of powers of metrically regular graphs. We prove that there is only one table of the parameters of an association scheme so that the corresponding metrically regular bipartite graph of diameter (7 distinct eigenvalues of the adjacency matrix) has the metrically regular square. The results deal with the graphs of the diameter see [7] and [8].
In this paper we consider the following problem: Over the class of all simple connected unicyclic graphs on vertices with girth (, being fixed), which graph minimizes the Laplacian spectral radius? Let be the lollipop graph obtained by appending a pendent vertex of a path on