We construct a graph complex calculating the integral homology of the bordered mapping class groups. We compute the homology of the bordered mapping class groups of the surfaces S1,1, S1,2 and S2,1. Using the circle action on this graph complex, we build a double complex and a spectral sequence converging to the homology of the unbordered mapping class groups. We compute the homology of the pun...

A sum graph is a graph for which there is a labeling of its vertices with positive integers so that two vertices are adjacent if and only if the sum of their labels is the label of another vertex. Integral sum graphs are defined similarly, except that the labels may be any integers. These concepts were first introduced by Harary, who provided examples of such graphs of all orders. The family of...

We consider four weighted matching-type problems: the bipartite graph and general graph versions of matching and f -factors. The linear program duals for these problems are shown to be weights of certain subgraphs. Specifically the so-called y duals are the weights of certain maximum matchings or f -factors; z duals (used for general graphs) are the weights of certain 2-factors or 2f -factors. ...

A join in a graph is a set F of edges such that for every circuit C, |C ∩ F | ≤ |C \ F |. We study the problem of finding a connected join covering a given subset of vertices of the graph, that is a Steiner tree which is a join at the same time. This turns out to contain the question of finding a T -join of minimum cardinality (or weight) which is, in addition, connected. This last problem is m...

Consider a graph whose edges have been colored red and blue. Assign a nonnegative real weight to every edge so that at every vertex, the sum of the weights of the incident red edges equals the sum of the weights of the incident blue edges. The set of all such assignments forms a convex polyhedral cone in the edge space, called the alternating cone. The integral (respectively, {0, 1}) vectors in...

In this paper, we develop two approaches to investigation of inverse spectral problems for a new class nonlocal operators on metric graphs. The Laplace differential operator is considered star-shaped graph with integral matching condition. This adjoint the functional-differential frozen argument at central vertex graph. We study problem that consists in recovery condition coefficients from eige...

In this paper, we investigate the strength of chromatic symmetric homology as a graph invariant. Chromatic is lift function for graphs to homological setting, and its Frobenius characteristic q,t generalization function. We exhibit three pairs where each pair has same but distinct over C Sn-modules. also show that integral contains torsion, based on computations, conjecture Z2-torsion in bigrad...

Let C = (V, E) be an undirected graph, w : E + Z’ a weight function and T c V an even subset of vertices from G. A T-cut is an edge-cut set which divides T into two odd sets. For ( Tj = 4 Seymour gave a good characterization of the graphs for which there exists a maximum packing of T-cuts that is integral. Based on Seymour’s characterization we provide a simple 0(n3m + PI”&) algorithm for incre...

We consider an Sp(4,Z) invariant expression involving two factors of the Kawazumi--Zhang (KZ) each which is a modular graph with one link, and four derivatives on moduli space genus Riemann surfaces. Manipulating it, we show that integral over linear combination links square KZ reduces to boundary integral. also three six space, from deduce In both cases, term completely determined by invariant...

A gain graph is a graph whose edges are orientably labelled from a group. A weighted gain graph is a gain graph with vertex weights from an abelian semigroup, where the gain group is lattice ordered and acts on the weight semigroup. For weighted gain graphs we establish basic properties and we present general dichromatic and forest-expansion polynomials that are Tutte invariants (they satisfy T...