For any finite, undirected, non-bipartite, vertex-transitive graph, we establish an explicit lower bound for the smallest eigenvalue of its normalised adjacency operator, which depends on graph only through degree and vertex-Cheeger constant. We also prove analogous result a large class irregular graphs, obtained as spanning subgraphs graphs. Using Babai, obtain operator in terms diameter degree.

The topic of this paper is the rectangle-free coloring of grids using four colors which is equivalent to the edge coloring of complete bipartite graphs without complete monochromatic subgraphs K2,2. Despite a strong mathematical background it is not known whether rectangle-free 4-colorable grids exist for five large grid sizes. We present in this paper an approach that solves the most complex p...

For every integer p > 0 let f(p) be the minimum possible value of the maximum weight of a cut in an integer weighted graph with total weight p. It is shown that for every large n and every m < n, f( ( n 2 ) + m) = b 2 4 c + min(d n 2 e, f(m)). This supplies the precise value of f(p) for many values of p including, e.g, all p = ( n 2 ) + ( m 2 ) when n is large enough and m 2 4 ≤ n 2 .

We study a Gaussian matrix function of the adjacency matrix of artificial and real-world networks. We motivate the use of this function on the basis of a dynamical process modeled by the time-dependent Schrödinger equation with a squared Hamiltonian. In particular, we study the Gaussian Estrada index-an index characterizing the importance of eigenvalues close to zero. This index accounts for th...

** This thesis studies graph clustering on bipartite and chordal graphs. It describes two types of clustering problems: a partition of the vertex of a graph with minimum diameters of subgraphs; maximizing possible number of edges given fixed number of vertices.

A graph is said to be non-singular if it has no eigenvalue equal to zero; otherwise it is singular. Molecular graphs that are non-singular and also have the property that all subgraphs of them obtained by deleting a single vertex are themselves singular, known as NSSD graphs, are of importance in the theory of molecular π-electron conductors; NSSD = non-singular graph with a singular deck. Wher...

In this paper we give optimal optimal vertex degree conditions that guarantee connection by short paths in ε-regular bipartite graphs. We also study a related question of decomposing an arbitrary graph into subgraphs of small diameter.

Suppose G is a graph with n vertices and m edges. Let n′ be the maximum number of vertices in an induced bipartite subgraph of G and let m′ be the maximum number of edges in a spanning bipartite subgraph of G. Then b(G) = m′/m is called the bipartite density of G, and b∗(G) = n′/n is called the bipartite ratio of G. This paper proves that every 2connected triangle-free subcubic graph, apart fro...

A new heuristic based on vertex invariants is developed to rapidly distinguish non-isomorphic graphs to a desired level of accuracy. The method is applied to sample subgraphs from an E.coli protein interaction network, and as a probe for discovery of extended motifs. The network’s structure is described using statistical properties of its N-node subgraphs for N ≤ 14. The Zipf plots for subgraph...

We study two extremal problems about subgraphs excluding a family F of graphs. i) Among all graphs with m edges, what is the smallest size f(m,F) of a largest F–free subgraph? ii) Among all graphs with minimum degree δ and maximum degree ∆, what is the smallest minimum degree h(δ,∆,F) of a spanning F– free subgraph with largest minimum degree? These questions are easy to answer for families not...