Signed graphs for portfolio analysis in risk management

We introduce the notion of structural balance for signed graphsin the context of portfolio analysis. A portfolio of securitiescan be represented as a signed graph with the nodes denotingthe securities and the edges representing the correlation betweenthe securities. With signed graphs, the characteristics of aportfolio from a risk management perspective can be uncoveredfor analysis purposes. It is shown that a portfolio characterizedby a signed graph of positive and negative edges that is structurallybalanced is characteristically more predictable. Investors whoundertake a portfolio position with all positively correlatedsecurities do so with the intention to speculate on the upside(or downside). If the portfolio consists of negative edges andis balanced, then it is likely that the position has a hedgingdisposition within it. On the other hand, an unbalanced signedgraph is representative of an investment portfolio which ischaracteristically unpredictable.     

The class reconstruction number of maximal planar graphs

The reconstruction numberrn(G) of a graphG was introduced by Harary and Plantholt as the smallest number of vertex-deleted subgraphsG _i =G – v _i in the deck ofG which do not all appear in the deck of any other graph. For any graph theoretic propertyP, Harary defined theP-reconstruction number of a graph G P as the smallest number of theG _i in the deck ofG, which do not all appear in the deck of any other graph inP We now study the maximal planar graph reconstruction numberrn(G), proving that its value is either 1 or 2 and characterizing those with value 1.     

On the corona of two graphs

Aequationes mathematicae -     

Paley graphs satisfy all first-order adjacency axioms

A graph satisfies Axiom n if, for any sequence of 2n of its points, there is another point adjacent to the first n and not to any of the last n. We show that, for each n, all sufficiently large Paley graphs satisfy Axiom n. From this we conclude at once that several properties of graphs are not first order, including self-complementarity and regularity.     

On double and multiple interval graphs

In this paper we discuss a generalization of the familiar concept of an interval graph that arises naturally in scheduling and allocation problems. We define the interval number of a graph G to be the smallest positive integer t for which there exists a function f which assigns to each vertex u of G a subset f(u) of the real line so that f(u) is the union of t closed intervals of the real line, and distinct vertices u and v in G are adjacent if and only if f(u) and f(v)meet. We show that (1) the interval number of a tree is at most two, and (2) the complete bipartite graph K_{m, n} has interval number ?(mn + 1)/(m + n)?.     

A graph theoretic approach to matrix inversion by partitioning

LetM be a square matrix whose entries are in some field. Our object is to find a permutation matrixP such thatPM P ^–1 is completely reduced, i.e., is partitioned in block triangular form, so that all submatrices below its diagonal are 0 and all diagonal submatrices are square and irreducible. LetA be the binary (0, 1) matrix obtained fromM by preserving the 0's ofM and replacing the nonzero entries ofM by 1's. ThenA may be regarded as the adjacency matrix of a directed graphD. CallD strongly connected orstrong if any two points ofD are mutually reachable by directed paths. Astrong component ofD is a maximal strong subgraph. Thecondensation D ^* ofD is that digraph whose points are the strong components ofD and whose lines are induced by those ofD. By known methods, we constructD ^* from the digraph,D whose adjacency matrixA was obtained from the original matrixM. LetA ^* be the adjacency matrix ofD ^*. It is easy to show that there exists a permutation matrixQ such thatQA ^* Q ^–1 is an upper triangular matrix. The determination of an appropriate permutation matrixP from this matrixQ is straightforward.This was an informal talk at the International Symposium on Matrix Computation sponsored by SIAM and held in Gatlinburg, Tennessee, April 24–28, 1961 and was an invited address at the SIAM meeting in Stillwater, Oklahoma on August 31, 1961     

The biparticity of a graph

The biparticity β(G) of a graph G is the minimum number of bipartite graphs required to cover G. It is proved that for any graph G, β(G) = {log₂χ(G)}. In view of the recent announcement of the Four Color Theorem, it follows that the biparticity of every planar graph is 2.