We consider simple labeled graphs, with non-zero labels in a ring. If the adjacency matrix of a labeled graph is invertible, the inverse matrix is a (labeled) adjacency matrix of another graph, called the inverse of the original graph. If the labeling takes place in an ordered ring, then balanced inverses—those with positive products of labels along every cycle—are of interest. We introduce the...

When the graph is a path with n vertices, we simply call PG the path polynomial and denote it by Pn. Define An as the adjacency matrix of a path on n vertices. For several interesting classes of graphs, A(Gi) is a polynomial in A(G), where Gi is the ith distance graph of G ([5]). Actually, for distance-regular graphs, A(Gi) is a polynomial in A(G) of degree i, and this property characterizes th...

Mathematical properties of extreme eigenfunctions of popular geographic weights matrices used in spatial statistics are explored, and applications of these properties are presented. Three theorems are proposed and proved. These theorems pertain to the popular binary geographic weights matrix––an adjacency matrix––based upon a planar graph. They uncover relationships between the determinant of t...

A uniform lift of a given network is a network with no loops and no multiple arrows that admits the first network as quotient. Given a regular network (in which all cells have the same type and receive the same number of inputs and all arrows have the same type) with loops or multiple arrows, we prove that it is always possible to construct a uniform lift whose adjacency matrix has only two pos...

An efficient algorithm leading to the Fries canonical structure is presented for benzenoid hydrocarbons. This is a purely topological approach, which is based on adjacency matrices and the Hadamard procedure of matrix multiplication. The idea is presented for naphthalene, as an example. The Fries canonical-structures are also derived for anthracene, coronene, triphenylene, phenanthrene, benz[a]...

Quantum algorithms for graph problems are considered, both in the adjacency matrix model and in an adjacency list-like array model. We give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, Strong Connectivity, Minimum Spanning Tree, and Single Source Shortest Paths. For example we show that the query complexity of Minimum Spanning Tree is in Θ(...