The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters. We consider the adjacency matrix of the ensemble of Erd˝ os-Rényi random graphs which consists of graphs on N vertices in which each edge occurs independently with probability p. We prove that in the regime pN ≫ 1, these matrices exhibit bulk universality in the sense tha...

Decomposing graphs into simpler graphs is one of the central concerns of graph theory. Investigations have revealed deep concepts such as modular decomposition, tree width or rank width, which measure—in different ways—the structural complexity of a graph’s topology. Courcelle and others have shown that such concepts can be used to obtain efficient algorithms for families of graphs that are ame...

LDPC codes of columnweight of two are constructed fromminimal distance graphs or cages. Distance graphs are used to represent LDPC code matrices such that graph vertices that represent rows and edges are columns. The conversion of a distance graph into matrix form produces an adjacency matrix with column weight of two and girth double that of the graph. The number of 1’s in each row (row weight...

Considering the Euclidean Jordan algebra of the real symmetric matrices endowed with the Jordan product and the inner product given by the usual trace of matrices, we construct an alternating Schur series with an element of the Jordan frame associated to the adjacency matrix of a strongly regular graph. From this series we establish necessary conditions for the existence of strongly regular gra...

In this paper, we describe the relationship between Perron root and eigenvectors of an irreducible subshift finite type with correlation forbidden words in subshift. particular, derive expression for associated adjacency matrix. As application, obtain (0,1) matrices which are directed graphs. Moreover, alternate definition Parry measure ergodic theory on type.

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...

Abstract: In this paper, we investigated relationships between the Fibonacci, Lucas, Padovan numbers and 1-factors of some bipartite graphs with upper Hessenberg adjacency matrix. We calculated permanent of these upper Hessenberg matrices by contraction method and show that their permanents are equal to elements of the Fibonacci, Lucas and Padovan numbers. At the end of the paper, we give some ...

Let G be a simple graph with adjacency matrix A(G) and π(G,x) the permanental polynomial of G. Let G × H denotes the Cartesian product of graphs G and H. Inspired by Klein’s idea to compute the permanent of some matrices (Mol. Phy., 1976, Vol. 31, (3): 811−823), in this paper in terms of some orientation of graphs we study the permanental polynomial of a type of graphs. Here are some of our mai...

This survey paper deals with upper and lower bounds on the number of k-matchings in regular graphs on N vertices. For the upper bounds we recall the upper matching conjecture which is known to hold for perfect matchings. For the lower bounds we first survey the known results for bipartite graphs, and their continuous versions as the van der Waerden and Tverberg permanent conjectures and its var...

This survey paper deals with upper and lower bounds on the number of k-matchings in regular graphs on N vertices. For the upper bounds we recall the upper matching conjecture which is known to hold for perfect matchings. For the lower bounds we first survey the known results for bipartite graphs, and their continuous versions as the van der Waerden and Tverberg permanent conjectures and its var...