Graph Theory

Magic Graphs by Alison M. Marr

By Alison M. Marr

Magic squares are one of the extra well known mathematical recreations. over the past 50 years, many generalizations of “magic” principles were utilized to graphs. lately there was a resurgence of curiosity in “magic labelings” because of a couple of effects that experience purposes to the matter of decomposing graphs into timber.

Key gains of this moment version include:

· a brand new bankruptcy on magic labeling of directed graphs

· purposes of theorems from graph thought and fascinating counting arguments

· new study difficulties and routines masking more than a few difficulties

· an absolutely up-to-date bibliography and index

This concise, self-contained exposition is exclusive in its specialise in the idea of magic graphs/labelings. it may possibly function a graduate or complicated undergraduate textual content for classes in arithmetic or desktop technological know-how, and as reference for the researcher.

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Extra resources for Magic Graphs

Sample text

Suppose G has vj vertices of degree j, for each i up to Δ, the largest degree represented in G. Then ke cannot be smaller than the sum obtained by applying the vΔ smallest labels to the vertices of degree Δ, the next-smallest values to the vertices of degree Δ − 1, and so on; in other words, 18 2 Edge-Magic Total Labelings Δ +(vΔ−1 ) ke ≥ (dΔ − 1)σ0vΔ + (dΔ−1 − 1)σvvΔ + ··· v +(v )+···+v Δ Δ−1 2 +σvΔ +(vΔ−1 )+···+v3 + v+e+1 . 2 An upper bound is achieved by giving the largest labels to the vertices of highest degree, and so on.

On the other hand, a low-energy pulse may be too weak to detect after reflection. The solution is to send a set of low-amplitude, narrowly defined pulses, and to increase the total energy when necessary by increasing the number of pulses. The most efficient way to proceed is to time the pulses in accordance with the marks on a ruler derived from a semi-graceful labeling. An array of detectors is distributed like a template of the transmitted pulse-train. The signal will match up precisely with the detectors when it returns.

An ) where n ≥ 4. Clearly 30 2 Edge-Magic Total Labelings ρ∗ (n) ≥ ρ(A) = an + an−1 − a2 − a1 + 1 = (an − a2 + 1) + (an−1 − a1 + 1) − 1 = σ(B) + σ(C) − 1 ≥ 2σ ∗ (n − 1) − 1. Moreover, equality can apply only if σ(B) = σ(C) = σ ∗ (n − 1). But σ(B) = σ(C) ⇒ an − a2 = an−1 − a1 ⇒ an−1 + a2 = an + a1 , which is impossible for a Sidon sequence A. Since σ ∗ and ρ∗ are integral, ρ∗ (n) ≥ 2σ ∗ (n − 1). 6 Prove that, when n ≥ 7, ρ∗ (n) ≥ n2 − 5n + 14. 15, ρ∗ (n) ≥ 2σ ∗ (n − 1) ≥ 2(4 + (n − 2)(n − 3) = n2 − 5n + 14.

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