This paper is based on a series of talks given at the Patejdlovka Enumeration Workshop held in the Czech Republic in November 2007. The topics covered are as follows. The graph polynomial, Tutte-Grothendieck invariants, an overview of relevant elementary finite Fourier analysis, the Tutte polynomial of a graph as a Hamming weight enumerator of its set of tensions (or flows), description of a fa...

The Shortest Cycle Cover Conjecture of Alon and Tarsi asserts that the edges of every bridgeless graph with m edges can be covered by cycles of total length at most 7m/5 = 1.400m. We show that every cubic bridgeless graph has a cycle cover of total length at most 34m/21 ≈ 1.619m and every bridgeless graph with minimum degree three has a cycle cover of total length at most 44m/27 ≈ 1.630m.

A recent strategy carried out by the aircraft industry to reduce the total cost of the new generation fighters has consisted in the development of a single airframe with different technical and operational specifications. This strategy has been designed to reduce costs in the Research, Design and Development phase with the ultimate objective of reducing the final unit price per aircraft. This i...

In estimating the cost-effectiveness of diagnostic procedures, it is helpful to treat diagnostic information as a commodity with a unit price. The amount of useful information provided by a test result can be measured in binary units (bits), and the unit price of the information produced by the test result can be expressed in dollars per bit in much the same way that the price of gold is given ...

We show that every plane graph of diameter 2r in which all inner faces are triangles and all inner vertices have degree larger than 5 can be covered with two balls of radius r : 2003 Wiley Periodicals, Inc. J Graph Theory 44:

The degree distance of a graph which is a degree analogue of the Wiener index of the graph. Let G = TUV C6[2p, q] be the carbon nanotubes covered by C6, formulas for calculating the degree distance of armchair polyhex nanotubes TUV C6[2p, q] are provided. Mathematics Subject Classification: 05C05, 05C12

Let Γ be a G-symmetric graph whose vertex set admits a nontrivial G-invariant partition B with block size v. Let ΓB be the quotient graph of Γ relative to B and Γ [B,C] the bipartite subgraph of Γ induced by adjacent blocks B,C of B. In this paper we study such graphs for which ΓB is connected, (G,2)-arc transitive and is almost covered by Γ in the sense that Γ [B,C] is a matching of v−1≥2 edge...

We call a vertex x of a graph G = (V,E) a codominated vertex if NG[y] ⊆ NG[x] for some vertex y ∈ V \{x}, and a graph G is called codismantlable if either it is an edgeless graph or it contains a codominated vertex x such that G − x is codismantlable. We show that (C4, C5)-free vertex-decomposable graphs are codismantlable, and prove that if G is a (C4, C5, C7)-free well-covered graph, then ver...

We show that for any 2-local colouring of the edges of a complete bipartite graph, its vertices can be covered with at most 3 disjoint monochromatic paths. And, we can cover almost all vertices of any complete or complete bipartite r-locally coloured graph with O(r) disjoint monochromatic cycles. MSC: 05C38, 05C55.

The following is proved: if every bridgeless graph G has a cycle cover of length at most 7/51€(G)I, then every bridgeless graph G has a cycle cover of length at most 7/51€(G)I such that any edge of G is covered once or twice.