Jerrold W. Grossman

Famous trails to Paul Erdős

The notion of Erd} os number has oated around the mathematical research community for more than thirty years, as a way to quantify the common knowledge that mathematical and scientiic research has become a very collaborative process in the twentieth century, not an activity engaged in solely by isolated individuals. In this paper we explore some (fairly short) collaboration paths that one can follow from Paul Erd} os to researchers inside and outside of mathematics. In particular, we nd that all the Fields Medalists up through 1998 have Erd} os numbers less than 6, and that over 60 Nobel Prize winners in physics, chemistry, economics, and medicine have Erd} os numbers less than 9. 1 An outstanding component of the collaboration graph The collaboration graph C has as vertices all researchers (dead or alive) from all academic disciplines, with an edge joining vertices u and v if u and v have jointly published a research paper or book (with possibly more co-authors). As is the case for any simple (undirected) graph, in C we have a notion of distance between two vertices u and v: d(u; v) is the number of edges in the shortest path between u and v, if such a path exits, 1 otherwise (it is understood that d(u; u) = 0). In this paper we are concerned with the collaboration subgraph centered at Paul Erd} os (1913{1996). For a researcher v, the number d(Paul Erd} os, v) is called

Generalized matrix tree theorem for mixed graphs

In this article we provide a combinatorial description of an arbitrary minor of the Laplacian matrix (L) of a mixed graph (a graph with some oriented and some unoriented edges). This is a generalized Matrix Tree Theorem. We also characterize the non-singular substructures of a mixed graph. The sign attached to a nonsingular substructure is described in terms of labeling and the number of unoriented edges included in certain paths. Nonsingular substructures may be viewed as generalized matchings, because in the case of disjoint vertex sets corresponding to the rows and columns of a minor of L, our generalized Matrix Tree Theorem provides a signed count over matchings between those vertex sets. A mixed graph is called quasi bipartite if it does not contain a non singular cycle (a cycle containing an odd number of un-oriented edges). We give several characterizations of quasi-bipartite graphs.

Fractional arboricity, strength, and principal partitions in graphs and matroids

Abstract In a 1983 paper, D. Gusfield introduced a function which is called (following W.H. Cunningham, 1985) the strength of a graph or matroid. In terms of a graph G with edge set E(G) and at least one link, this is the function η(G) = minF⊆E(G) ∣F∣/(ω(G − F) − ω(G)), where the minimum is taken over all subsets F of E(G) such that ω(G − F), the number of components of G − F, is at least ω(G) + 1. In a 1986 paper, C. Payan introduced the fractional arboricity of a graph or matroid. In terms of a graph G with edge set E(G) and at least one link this function is γ(G) = maxH⊆G ∣E(H)∣/(∣V(H)∣ − ω(H)), where H runs over all subgraphs of G having at least one link. Connected graphs G for which γ(G) = η(G) were used by A. Rucinski and A. Vince in 1986 while studying random graphs. We characterize the graphs and matroids G for which γ(G) = η(G). The values of γ and η are computed for certain graphs, and a recent result of Erdos (that if each edge of G lies in a C3, then ∣E(G)∣≥ 3 2 (∣V(G)∣ − 1)) is generalized in terms of η. The principal partition of a graph was introduced in 1967 by G. Kishi and Y. Kajitani, by T. Ohtsuki, Y. Ishizaki, and H. Watanabe, and by M. Iri (all of these were published in 1968). It has been used since then for the analysis of electrical networks in which the two Kirchhoff laws and Ohms law hold, because it often allows the currents and voltage drops in the network to be completely computed with fewer measurements than are required for either of the Kirchhoff laws used alone. J. Bruno and L. Weinberg generalized the principal partition to matroids in 1971, and their generalization was refined independently by N. Tomizawa (1976) and by H. Narayanan and M.N. Vartak (1974, 1981). Here we demonstrate that γ and η are closely related to the principal partition and can be used to give a simple definition of both the principal partition and the more recent refinements of it.

Algebraic graph theory without orientation

Abstract Let G be an undirected graph with vertices {v 1 ,v 2 ,…,>;v ⋎ } and edges {e1,e2, …,eϵ}. Let M be the ⋎ × ϵ matrix whose ijth entry is 1 if ej is a link incident with vi, 2 if ej is a loop at vi, and 0 otherwise. The matrix obtained by orienting the edges of a loopless graph G (i.e., changing one of the 1s to a − 1 in each column of M) has been studied extensively in the literature. The purpose of this paper is to explore the substructures of G and the vector spaces associated with the matrix M without imposing such an orientation. We describe explicitly bases for the kernel and range of the linear transformation from Rϵ to R⋎ defined by M. Our main results are determinantal formulas, using the unoriented Laplacian matrix MMt, to count certain spanning substructures of G. These formulas may be viewed as generalizations of the matrix tree theorem. The point of view adopted in this paper also gives rise to a matroid structure on the edges of G analogous to the cycle matroid and its dual. In this setting, the analogue of a spanning forest can have components with one odd cycle, and the analogue of an edge cut has the property that its removal creates a new bipartite component.

On the minors of an incidence matrix and its Smith normal form

Abstract Consider the vertex-edge incidence matrix of an arbitrary undirected, loopless graph. We completely determine the possible minors of such a matrix. These depend on the maximum number of vertex-disjoint odd cycles (i.e., the odd tulgeity ) of the graph. The problem of determining this number is shown to be NP-hard. Turning to maximal minors, we determine the rank of the incidence matrix. This depends on the number of components of the graph containing no odd cycle. We then determine the maximum and minimum absolute values of the maximal minors of the incidence matrix, as well as its Smith normal form. These results are used to obtain sufficient conditions for relaxing the integrality constraints in integer linear programming problems related to undirected graphs. Finally, we give a sufficient condition for a system of equations (whose coefficient matrix is an incidence matrix) to admit an integer solution.

Alternating cycles in edge-partitioned graphs

Abstract It is shown that if the edges of a 2-connected graph G are partitioned into two classes so that every vertex is incident with edges from both classes, then G has an alternating cycle. The connectivity assumption can be dropped if both subgraphs resulting from the partition are regular, or have only vertices of odd degree.

Edge version of the matrix tree theorem for trees

We provide a combinatorial description of all the minors of the edge version of the Laplacian matrix of a mixed tree. The description involves the common SDRs for the forests obtained by deleting from the tree the edge sets corresponding to the row and column indices of the minor.

Distance formula and shortest paths for the (n,k)-star graphs

The class of (n,k)-star graphs is a generalization of the class of star graphs. Thus a distance formula for the first class implies one for the second. In this paper, we show that the converse is also true. Another important concept is the number of shortest paths between two vertices. This problem has been solved for the star graphs. We will solve the corresponding problem for the (n,k)-star graphs.

Time-stamped graphs and their associated influence digraphs

A time-stamped graph is an undirected graph with a real number on each edge. Vertex u influences vertex v if there is a non-decreasing path from u to v. The associated influence digraph of a time-stamped graph is the directed graph that records the influences. Among other results, we determine for what n and t there exists a time-stamped graph whose associated influence digraph has n vertices and t arcs. We also investigate the minimum number of vertices a graph can have so that a given digraph is an induced subgraph of its associated influence digraph. A number of other questions are also explored.

Generalized ramsey theory for graphs, x

The double star S(n, m), where n Â? m Â? 0, is the graph consisting of the union of two stars K1,n and K1,m together with a line joining their centers. Its ramsey number r(S(n, m)) is the least number p such that there is a monochromatic copy of S(n, m) in any 2-coloring of the edges of Kp. It is shown that r(S(n, m)) = max (2n + 1, n + 2m + 2) if n is odd and mÂ?2 and r(S(n, m)) = max (2n + 2, n + 2m + 2) otherwise, for n Â? Â?2morn Â? 3m.