We take G to be an undirected graph without loops or multiple edges, with vertex set V (G) = f1; : : : ; ng, and with (0; 1)-adjacency matrix A. Let P denote the orthogonal projection of IR onto the eigenspace E( ) of A, and let fe1; : : : ; eng be the standard orthonormal basis of IR. Since E( ) is spanned by the vectors Pej (j = 1; : : : ; n) there exists X V (G) such that the vectors Pej (j ...

Let G be a finite graph with μ as an eigenvalue of multiplicity k. A star set for μ is a set X of k vertices in G such that μ is not an eigenvalue of G−X. We investigate independent star sets of largest possible size in a variety of situations. We note connections with symmetric designs, codes, strongly regular graphs, and graphs with least eigenvalue −2. AMS Classification: 05C50

let $g$ be a simple graph with vertex set $v(g) = {v_1, v_2,ldots, v_n}$ and $d_i$ the degree of its vertex $v_i$, $i = 1, 2,cdots, n$. inspired by the randi'c matrix and the general randi'cindex of a graph, we introduce the concept of general randi'cmatrix $textbf{r}_alpha$ of $g$, which is defined by$(textbf{r}_alpha)_{i,j}=(d_id_j)^alpha$ if $v_i$ and $v_j$ areadjacent, and zero otherwise. s...

A graph G with convex-QP stability number (or simply a convex-QP graph) is a graph for which the stability number is equal to the optimal value of a convex quadratic program, say P (G). There are polynomial-time procedures to recognize convex-QP graphs, except when the graph G is adverse or contains an adverse subgraph (that is, a non complete graph, without isolated vertices, such that the lea...

In Lecture 10, we introduced a fundamental object of spectral graph theory: the graph Laplacian, and established some of its basic properties. We then focused on the task of estimating the value of eigenvalues of Laplacians. In particular, we proved the Courant-Fisher theorem that is instrumental in obtaining upper-bounding estimates on eigenvalues. Today, we continue by showing a technique – s...

Recently some important results have been proved showing that the gap between the largest eigenvalue A: of a finite regular graph of valency k and its second eigenvalue is related to expansion properties of the graph [1]. In this paper we investigate infinite graphs and show that in this case the expansion properties are related to the spectral radius of the graph. First we introduce necessary ...

The infimum of the least eigenvalues of all finite induced subgraphs of an infinite graph is defined to be its least eigenvalue. In [P.J. Cameron, J.M. Goethals, J.J. Seidel and E.E. Shult, Line graphs, root systems, and elliptic geometry, J. Algebra 43 (1976) 305–327], the class of all finite graphs whose least eigenvalues > −2 has been classified: (1) If a (finite) graph is connected and its ...

The algebraic connectivity of a graph, which is the second-smallest eigenvalue of the Laplacian of the graph, is a measure of connectivity. We show that the problem of adding a specified number of edges to an input graph to maximize the algebraic connectivity of the augmented graph is NP-hard.

Abstract Edges in the graph associated with a square matrix over field may be classified as to how their removal affects multiplicity of an identified eigenvalue. There are five possibilities: + 2 +2 (2-Parter); 1 +1 (Parter); no change (neutral); − -1 (downer); and -2 (2-downer)....