A matrix representation of graphs and its spectrum as a graph invariant

We use the line digraph construction to associate an orthogonal matrix with each graph. From this orthogonal matrix, we derive two further matrices. The spectrum of each of these three matrices is considered as a graph invariant. For the first two cases, we compute the spectrum explicitly and show that it is determined by the spectrum of the adjacency matrix of the original graph. We then show by computation that the isomorphism classes of many known families of strongly regular graphs (up to 64 vertices) are characterized by the spectrum of this matrix. We conjecture that this is always the case for strongly regular graphs and we show that the conjecture is not valid for general graphs. We verify that the smallest regular graphs which are not distinguished with our method are on 14 vertices.Comment: 14 page

The need for church-sponsored programs for the mentally retarded and their availability in Fayette and Jessamine counties (Kentucky)

https://place.asburyseminary.edu/ecommonsatsdissertations/2175/thumbnail.jp

The impact of Ceasar Augustus on Roman trade

This paper studies Gaius Julius Octavian Caesar (Augustus) who combined the military/political expertise of a Roman Patrician with the practical business sense of the upper middle class (Equites) and as a consequence, his administration revived Rome’s economic fortunes and launched a new period of economic prosperity for the Empire. It also explores the specific actions of Augustus and those near to him that led directly to the revival of Rome’s economic fortunes and the remodeling of Rome and its many celebrated monuments and essential buildings. It concludes with a final summation of the man and his contributions to Rome’s success as an empire and a powerful force in the known world during this time period

Discoveries in New Testament texts since nineteen hundred

Thesis (M.A.)--Boston University, 194

Pattern vectors from algebraic graph theory

Graphstructures have proven computationally cumbersome for pattern analysis. The reason for this is that, before graphs can be converted to pattern vectors, correspondences must be established between the nodes of structures which are potentially of different size. To overcome this problem, in this paper, we turn to the spectral decomposition of the Laplacian matrix. We show how the elements of the spectral matrix for the Laplacian can be used to construct symmetric polynomials that are permutation invariants. The coefficients of these polynomials can be used as graph features which can be encoded in a vectorial manner. We extend this representation to graphs in which there are unary attributes on the nodes and binary attributes on the edges by using the spectral decomposition of a Hermitian property matrix that can be viewed as a complex analogue of the Laplacian. To embed the graphs in a pattern space, we explore whether the vectors of invariants can be embedded in a low- dimensional space using a number of alternative strategies, including principal components analysis ( PCA), multidimensional scaling ( MDS), and locality preserving projection ( LPP). Experimentally, we demonstrate that the embeddings result in well- defined graph clusters. Our experiments with the spectral representation involve both synthetic and real- world data. The experiments with synthetic data demonstrate that the distances between spectral feature vectors can be used to discriminate between graphs on the basis of their structure. The real- world experiments show that the method can be used to locate clusters of graphs