A drawing of a graph in the plane is even if nonadjacent edges have an even number of intersections. Hanani’s theorem characterizes planar graphs as those graphs that have an even drawing. In this paper we present an algebraic characterization of graphs that have an even drawing. Together with Hanani’s theorem this yields an algebraic characterization of planar graphs. We will also present alge...

If G is a finite graph, a proper coloring of G is a way to color the vertices of the graph using n colors so that no two vertices connected by an edge have the same color. (The celebrated four-color theorem asserts that if G is planar, then there is at least one proper coloring of G with four colors.) By a classical result of Birkhoff, the number of proper colorings of G with n colors is a poly...

A classical difficult isomorphism testing problem is to test isomorphism of p-groups of class 2 and exponent p in time polynomial in the group order. It is known that this problem can be reduced to solving the alternating matrix space isometry problem over a finite field in time polynomial in the underlying vector space size. We propose a venue of attack for the latter problem by viewing it as ...

We study three mixing properties of a graph: large algebraic connectivity, large Cheeger constant (isoperimetric number) and large spectral gap from 1 for the second largest eigenvalue of the transition probability matrix of the random walk on the graph. We prove equivalence of this properties (in some sense). We give estimates for the probability for a random graph to satisfy these properties....

Spectral graph theory looks at the interplay between the structure of a graph and the eigenvalues of a matrix associated with the graph. Many interesting graphs have rich structure which can help in determining the eigenvalues associated with some particular matrix of a graph. This survey looks at some common techniques in working with and determining the eigenvalues associated with the normali...

Hierarchically distributed graph transformation ooers means to model diierent aspects of open distributed systems very intuitively in a graphical way. The distribution topology as well as local object structures are represented graphically. Distributed actions such as local actions, network activities, communication and synchronization can be described homogeneously using the same method: graph...

A balanced graph is a bipartite graph with no induced circuit of length 2 (mod 4). These graphs arise in linear programming. We focus on graph-algebraic properties of balanced graphs to prove a complete classification of balanced Cayley graphs on abelian groups. Moreover, in Section 5 of this paper, we prove that there is no cubic balanced planar graph. Finally, some remarkable conjectures for ...

Given a perfect coloring of graph, we prove that the $L_1$ distance between two rows adjacency matrix graph is not less than corresponding parameter coloring. With help an algebraic approach, deduce corollaries this result for $2$-colorings, colorings in distance-$l$ graphs and distance-regular graphs. We also provide examples when obtained property reject several putative matrices infinite

We present an extremely fast graph drawing algorithm for very large graphs, which we term ACE (for Algebraic multigrid Computation of Eigenvectors). ACE exhibits an improvement of something like two orders of magnitude over the fastest algorithms we are aware of; it draws graphs of millions of nodes in less than a minute. ACE finds an optimal drawing by minimizing a quadratic energy function. T...

Let G be a weighted graph. Let v be a vertex of G and let Gω denote the graph obtained by adding a vertex u and an edge {v, u} with weight ω to G. Then the algebraic connectivity μ(Gω) of G v ω is a nondecreasing function of ω and is bounded by the algebraic connectivity μ(G) of G. The question of when lim ω→∞ v ω) is equal to μ(G) is considered and answered in the case that G is a tree.