Hoede-sequences

In an attempt to prove the double-cycle-conjecture for cubic graphs,\ud C. Hoede formulated the following combinatorial problem.\ud “Given a partition of {1, 2, . . . , 3n} into n equal classes, is\ud it possible to choose from each class a number such that\ud these numbers form an increasing sequence of alternating\ud parity?U+00e2U+0080?\ud Let a Hoede-sequence be defined as an increasing sequence of natural\ud numbers of alternating parity. We determine the average number of\ud Hoede-sequences w.r.t. arbitrary partitions, and obtain bounds for the\ud maximum and minimum number of Hoede-sequences w.r.t. partitions\ud into equal classes.\u

How can the Power of Leviathans be Measured?

In certain respects, it seems expedient to describe a government as a homogeneous and self-interested entity, called ’Leviathan’. To optimize fiscal constraints, we need to know how powerful a Leviathan really is. This paper presents a new approach to measure the power of Leviathans. This new approach defines fiscal fiscal power in terms of income deviation. It supposes that there exists a positive connection between fiscal power and intergovernmental grants. To examine the approach empirically, we use data on U.S. counties in the period 1999-2002. Equations of fiscal power are estimated on the full and on stratified samples. Overall, the results support the new approach. Nonetheless, further research on the highly significant control variables would be needed to derive recommendations for more efficient fiscal constraints.Leviathan; measurement; income deviation; grants

Even graphs

A nontrivial connected graph G is called even if for each vertex v of G there is a unique vertex [bar v] such that d(v,[bar v]) = diam G. Special classes of even graphs are defined and compared to each other. In particular, an even graph G is called symmetric if d(u,v) + d(u,[bar v]) = diam G for all u, v V(G). Several properties of even and symmetric even graphs are stated. For an even graph of order n and diameter d other than an even cycle it is shown that n ≥ 3d - 1 and conjectured that n ≥ 4d - 4. This conjecture is proved for symmetric even graphs and it is shown that for each pair of integers n, d with n even, d ≥ 2 and n ≥ 4d - 4 there exists an even graph of order n and diameter d. Several ways of constructing new even graphs from known ones are presented

How can the power of Leviathans be measured?

In certain respects, it seems expedient to describe a government as a homogeneous and self-interested entity, called ’Leviathan’. To optimize fiscal constraints, we need to know how powerful a Leviathan really is. This paper presents a new approach to measure the power of Leviathans. This new approach defines fiscal power in terms of income deviation. It supposes that there exists a positive connection between fiscal power and intergovernmental grants. To examine the approach empirically, we use data on U.S. counties in the period 1999-2002. Equations of fiscal power are estimated on the full and on stratified samples. Overall, the results support the new approach. Nonetheless, further research on the highly significant control variables would be needed to derive recommendations for more efficient fiscal constraints.Leviathan; measurement; income deviation; grants