This quantity offers easy-to-understand but stunning homes acquired utilizing topological, geometric and graph theoretic instruments within the parts lined through the Geometry convention that happened in Mulhouse, France from September 7–11, 2014 in honour of Tudor Zamfirescu at the get together of his seventieth anniversary. The contributions tackle matters in convexity and discrete geometry, in distance geometry or with geometrical taste in combinatorics, graph idea or non-linear research. Written by way of most sensible specialists, those papers spotlight the shut connections among those fields, in addition to ties to different domain names of geometry and their reciprocal impact. they provide an outline on fresh advancements in geometry and its border with discrete arithmetic, and supply solutions to a number of open questions. the amount addresses a wide viewers in arithmetic, together with researchers and graduate scholars drawn to geometry and geometrical difficulties.

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**Example text**

It was something against any standard scenario. 43 “I Am Not a Serious Mathematician” This was the beginning of his recent message to me (July 3, 2014). And he added: “…and saying this, I am serious. I don’t deserve so much attention from your part. Other mathematicians work very hard. It is not my case. Doing math means, for me, to play. I enjoy beautiful, but unexpected, surprising things, but such things are sometimes not so difficult. For instance, my aim now is to fix a convex body by a circle avoiding its interior.

Therefore the corresponding multiset of parts, A⊗ , has size |A⊗ | = j m j ≥ 2. Consequently, the OGF of the corresponding conumerants c(n) equals c(n)x n = (x) := n≥0 1 Q(x) (4) where the denominator ν Q(x) = 1 − q(x) = 1 − m j xaj (5) j=1 is an important polynomial which we call (after [24]) the co-characteristic polynomial of the OGF. Since Q(0) = 1, all roots of Q(x) are nonzero. Since Q(x) is decreasing for real x > 0 (because then Q (x) < 0) and Q(1) = 1 − |A⊗ | < 0, Q(x) has a unique (simple) real positive root, say τ , such that 0 < τ < 1.

For instance, the subgraph induced by the vertex set {g, gg, gf } in T (D0 ) in our Figure forms one GBS, while in S(D) and M(D) the same subgraph must be covered by two GBSs, due to the lack of the loop at the vertex g in S(D) and M(D). The problem of finding equivalent results for other transformations of digraphs, such as various power digraphs, remains also open. References 1. J. Akiyama, T. Hamada, I. Yoshimura, On characterizations of the middle graphs. TRU Math. 11, 35–39 (1975) 2. M. Behzad, A criterium for the planarity of the total graph.