By Daniel Abraham
The robust city-state of Saraykeht is a bastion of peace and tradition, a massive heart of trade and alternate. Its economic climate is determined by the facility of the captive spirit, Seedless, an andat guaranteed to the poet-sorcerer Heshai for all times. input the Galts, a juggernaut of an empire devoted to laying waste to all lands with their ferocious military. Saraykeht, although, has regularly been too powerful for the Galts to assault, yet now they see a chance. in the event that they can put off Heshai, Seedless’s bonded poet-sorcerer, Seedless will perish and the whole urban will fall. With mystery forces contained in the urban, the Galts organize to enact their bad plan.
In the center is Otah, an easy laborer with a posh prior. Recruited to behave as a bodyguard for his girlfriend's boss at a mystery assembly, he inadvertently learns of the Galtish plot. Otah reveals himself because the sole desire of Saraykeht, both he stops the Galts, or the complete urban and everybody in it perishes ceaselessly.
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Additional info for A Shadow in Summer (Long Price Quartet, Book 1)
1 Assume that T is a tree that admits an ˛-valuation f and let A and B be its stable sets. Assume that jAj jBj. Without loss of restriction, we can also assume that the characteristic of f is assigned to jAj. v/ which verifies this condition. 3; 2/-EAV labelings of trees. 3 () Let T be tree that admits an ˛-valuation with stable sets A and B. 3; 2/-EAV labeling. Proof Let f be an ˛-valuation of a tree T of order p with characteristic k. 1, we can assume that jAj jBj and that the vertex labeled k belongs to A.
V//=2; if v 2 B: Thus, f assigns to the vertices of A the labels Œ0; dp=2e 1 and to the vertices of B the labels Œdp=2e; p 1. y//=2. T/g D Œ1; p 1, which implies that f is a graceful labeling of T. Since the labels of the vertices in A are less than the labels of the vertices in B, f is an ˛-valuation of T. This proves the result. 4. Using the two previous results the following theorem can be proved. 3; 2/-EAV labeling if and only if T admits an ˛-valuation and jjAj jBjj Ä 1, where A and B are the stable sets of T.
So far, this is still an open question and Enomoto et al. proposed the following conjecture, that has become very popular. 1 () All trees are super edge-magic. 2). From this point on, we will concentrate on the edge-magicness and the super edge-magicness of some specific families of graphs, and about what can be said about them. We already know that the star K1;n is super edge-magic, since it is a particular family of caterpillar. Moreover, we have the following result. 3 () There are exactly 3 2n distinct edge-magic labelings of the star K1;n , of which only two are super edge-magic, up to isomorphisms.