We prove the following Turán-Type result: If there are more than 9mn/16 edges in a simple and bipartite Eulerian digraph with vertex partition size m and n, then the graph contains a directed cycle of length 4 or 6. By using this result, we improve an upper bound for the diameter of interchange graphs.

Phil Hanlon proved that the coefficients of the chromatic polynomial of a graph G are equal (up to sign) to the dimensions of the summands in a Hodge-type decomposition of the top homology of the coloring complex for G. We prove a type B analogue of this result for chromatic polynomials of signed graphs using hyperoctahedral Eulerian idempotents.

Abstract It is well-known that every Eulerian orientation of an 2 k -edge-connected undirected graph -arc-connected. A long-standing goal in the area has been to obtain analogous results for vertex-connectivity. Levit, Chandran and Cheriyan recently proved Levit et al. (2018) a hypercube dimension -vertex-connected. Here we provide elementary proof this result. We also show other families -regu...

The Wiener index of a connected graph is the sum distances between all unordered pairs vertices. A Eulerian if its vertex degrees are even. In Gutman et al. (2014) authors proved that cycle unique maximising among graphs given order. They also conjectured for order n≥26 consisting on n−2 vertices and triangle share with second largest index. conjecture known to hold n≤25 exception six values. t...

A recent series of breakthroughs initiated by Spielman and Teng culminated in the construction of nearly linear time Laplacian solvers, approximating the solution of a linear system Lx = b, where L is the normalized Laplacian of an undirected graph. In this paper we study the space complexity of the problem. Surprisingly we are able to show a probabilistic, logspace algorithm solving the proble...

Counting problems are studied in a variety of areas. For example, enumerative combinatorics, statistics, statistical physics, and artificial intelligence. In this dissertation, we investigate several counting problems, which are subjects of active research. The specific problems considered are: counting independent sets in bipartite graphs (#BIS), computing the partition function of the hard-co...