Nonsingular Graph Research Articles

Graphs whose adjacency matrices have rank equal to the number of distinct nonzero rows

For a simple graph G, let rank(G) and dnzr(G) denote respectively the rank and the number of distinct nonzero rows of the adjacency matrix A(G) of G. Equivalent conditions are given for the join G1∨G2 of two vertex-disjoint graphs G1,G2 to satisfy rank(G1∨G2)=dnzr(G1∨G2). A new proof is provided for the known relation rank(G)=dnzr(G) for cographs G. Our approach relies on the concepts of neighborhood equivalence classes, reduced graph and reduced adjacency matrix, and also on a known result that relates the spectrum of the adjacency matrix of a graph with that of its reduced adjacency matrix as well as a new characterization of the nonsingularity of a real symmetric matrix in a special 2×2 block form. Our treatment provides ways to construct graphs G, other than cographs, that satisfy rank(G)=dnzr(G). As a side result we also show that every rational number is equal to the sum of the entries of the inverse of the adjacency matrix of a connected nonsingular graph.

Graphs with a common eigenvalue deck

In this paper, we consider graphs whose deck consists of cards (which are the vertex-deleted subgraphs) that share the same eigenvalue, say μ . We show that, the characteristic polynomial can be reconstructed from the deck, providing a new proof of Tutte’s result for this class of graphs. Moreover, for the subclass of non-singular graphs, the graph can be uniquely reconstructed from the eigenvectors of the cards associated with the eigenvalue μ . The remaining graphs in this class are shown to be μ -cores graphs.

Nonsingular mixed graphs with few eigenvalues greater than two

Let G be a nonsingular connected mixed graph. We determine the mixed graphs G on at least seven vertices with exactly two Laplacian eigenvalues greater than 2. In addition, all mixed graphs G with exactly one Laplacian eigenvalue greater than 2 are also characterized.

Sign-nonsingular skew-symmetric matrices

Sign-nonsingular skew-symmetric matrices are investigated. The concept of a nonsingular graph is introduced, and such graphs are related to pfaffian graphs. Considerable attention is devoted to properties of sign-nonsingular skew-symmetric matrices A = ( a ij ) for which there do not exist sign-nonsingular skew-symmetric matrices B = ( b ij ) of the same order with more nonzero entries and a ij = 0 whenever b ij = 0.

A short algorithm to delete singular junctions in bond graphs

Singular junctions, singular and nonsingular bond graphs are defined. Singular and nonsingular causal assignments, consistent and inconsistent sets of input variables, and the concept of precausal equivalence are also defined. Nonsingular bond graphs are important because they are precisely the bond graphs in which consistent causal assignments correspond to nonsingular causal assignments. A simple, polynomial time algorithm is described that constructs from a given bond graph a precausally equivalent nonsingular bond graph by deleting a set of singular junctions. This algorithm simplifies modelling and makes it easier to find a set of state space equations.

Application of graphs to the Gaussian elimination method

Certain algebraic operations (in the Boolean sense) are developed for directed graphs. Nonsingular and inverse graphs are defined and some of their characteristics are derived. The results are applied for the Gaussian elimination process.