Singular values of tournament matrices.
This paper contains a number of results in the theory of star partitions of graphs. We illustrate a variety of situations which can arise when the Reconstruction Theorem for graphs is used, considering in particular galaxy graphs - these are graphs in which every star set is independent. We discuss a recursive ordering of graphs based on the Reconstruction Theorem, and point out the significance of galaxy graphs in this connection.
Let be the wheel graph on vertices, and let be the graph on vertices obtained by attaching pendant edges together with hanging paths of length two at vertex , where is the unique common vertex of triangles. In this paper we show that (, ) and are determined by their signless Laplacian spectra, respectively. Moreover, we also prove that and its complement graph are determined by their Laplacian spectra, respectively, for and .
In this paper we derive new properties complementary to an Brualdi-Li tournament matrix . We show that has exactly one positive real eigenvalue and one negative real eigenvalue and, as a by-product, reprove that every Brualdi-Li matrix has distinct eigenvalues. We then bound the partial sums of the real parts and the imaginary parts of its eigenvalues. The inverse of is also determined. Related results obtained in previous articles are proven to be corollaries.
Let be a simple connected graph with vertex set and edge set , and let be the degree of the vertex . Let be the distance matrix and let be the diagonal matrix of the vertex transmissions of . The generalized distance matrix of is defined as , where . Let be the generalized distance eigenvalues of , and let be an integer with . We denote by the sum of the largest generalized distance eigenvalues. The generalized distance spread of a graph is defined as . We obtain some...
The distance Laplacian of a connected graph is defined by , where is the distance matrix of , and is the diagonal matrix whose main entries are the vertex transmissions in . The spectrum of is called the distance Laplacian spectrum of . In the present paper, we investigate some particular distance Laplacian eigenvalues. Among other results, we show that the complete graph is the unique graph with only two distinct distance Laplacian eigenvalues. We establish some properties of the distance...