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For a connected graph G of order p and a set S ⊆ V (G) , the Steiner distance of S is the minimum number of edges in a connected subgraph of G containing S. If n is an integer, 2 ≤ n ≤ p and a vertex v ∈ V (G) , the Steiner n -eccentricity of a vertex v of G , ex n (v) , is the maximum Steiner distance of all n -subsets of V (G) containing v. The Steiner n -radius of G , rad n (G) , is the minimum Steiner n -eccentricities of all vertices in G. We give bounds on rad n (G) in terms of the order of G and the minimum degree of G for all graphs and for graphs that contain no triangles. We shall also investigate the relation between the n -radius of a graph G and its complement Ḡ. [ABSTRACT FROM AUTHOR]

Let S be a set of vertices of a connected graph G. The Steiner distance of S is the minimum size among all connected subgraphs of G containing the vertices of S. The sum of all Steiner distances on sets of size k is called the Steiner k-Wiener index. We obtain formulae for bounds on the Steiner–Wiener index of graphs in terms of chromatic number, vertex connectivity and independence number. Also, we obtain bounds for 4-cycle free graphs. [ABSTRACT FROM AUTHOR]

For a vertex set S of G , we use κ S (G) to denote the maximum number of edge-disjoint Steiner trees of G such that any two of such trees intersect in S. The generalized k -connectivity of G is defined as κ k (G) = min S ⊆ V (G) , | S | = k κ S (G). We get that for any generalized Petersen graph G = G P (n , k) with n ≥ k + 3 , κ t (G) = 1 when t ≥ 2 k + 5. We give the values of κ k (G) for Petersen graph G P (5 , 2) , where 3 ≤ k ≤ 1 0 , and the values of κ k (G) for generalized Petersen graph G P (n , 1) , where n ≥ 3 and 3 ≤ k ≤ 2 n. [ABSTRACT FROM AUTHOR]

Given a spectral triple on a C ∗ -algebra together with a unital injective endomorphism α , the problem of defining a suitable crossed product C ∗ -algebra endowed with a spectral triple is addressed. The proposed construction is mainly based on the works of Cuntz and [A. Hawkins, A. Skalski, S. White and J. Zacharias, On spectral triples on crossed products arising from equicontinuous actions, Math. Scand. 113(2) (2013) 262–291], and on our previous papers [V. Aiello, D. Guido and T. Isola, Spectral triples for noncommutative solenoidal spaces from self-coverings, J. Math. Anal. Appl. 448(2) (2017) 1378–1412; V. Aiello, D. Guido and T. Isola, A spectral triple for a solenoid based on the Sierpinski gasket, SIGMA Symmetry Integrability Geom. Methods Appl. 17(20) (2021) 21]. The embedding of α () in can be considered as the dual form of a covering projection between noncommutative spaces. A main assumption is the expansiveness of the endomorphism, which takes the form of the local isometricity of the covering projection, and is expressed via the compatibility of the Lip-norms on and α (). [ABSTRACT FROM AUTHOR]

A Steiner triple system (STS) is a set S together with a collection ℬ of subsets of S of size 3 such that any two elements of S belong to exactly one element of ℬ. It is well known that the class of finite STS has a Fraïssé limit M F . Here, we show that the theory T Sq ∗ of M F is the model completion of the theory of STSs. We also prove that T Sq ∗ is not small and it has quantifier elimination, TP 2 , NSOP 1 , elimination of hyperimaginaries and weak elimination of imaginaries. [ABSTRACT FROM AUTHOR]

In this paper, the Sphere-packing bound, Wang-Xing-Safavi-Naini bound, Johnson bound and Gilbert-Varshamov bound on the subspace code of length 2 ν + δ , size M , minimum subspace distance 2 j based on m -dimensional totally singular subspace in the (2 ν + δ) -dimensional orthogonal space 𝔽 q (2 ν + δ) over finite fields 𝔽 q of characteristic 2, denoted by (2 ν + δ , M , 2 j , m) q , are presented, where ν is a positive integer, δ = 0 , 1 , 2 , 0 ≤ m ≤ ν , 0 ≤ j ≤ m. Then, we prove that (2 ν + δ , M , 2 j , m) q codes attain the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures in ℳ (m , 0 , 0 ; 2 ν + δ) , where ℳ (m , 0 , 0 ; 2 ν + δ) denotes the collection of all the m -dimensional totally singular subspaces in the (2 ν + δ) -dimensional orthogonal space 𝔽 q (2 ν + δ) over 𝔽 q of characteristic 2. Finally, Gilbert-Varshamov bound and linear programming bound on the subspace code (2 ν + δ , M , d) q in ℳ (2 ν + δ) are provided, where ℳ (2 ν + δ) denotes the collection of all the totally singular subspaces in the (2 ν + δ) -dimensional orthogonal space 𝔽 q (2 ν + δ) over 𝔽 q of characteristic 2. [ABSTRACT FROM AUTHOR]

The Steiner distance of a graph, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the concept of classical graph distance. For a connected graph G of order at least 2 and , the Steiner distance dG(S) among the vertices of S is the minimum size among all connected subgraphs whose vertex sets contain S. Let n, k be two integers with 2 ≤ k ≤ n. Then the Steiner k-eccentricity ek(v) of a vertex v of G is defined by . Furthermore, the Steiner k-diameter of G is . In 2011, Chartrand, Okamoto and Zhang showed that k − 1 ≤ sdiamk(G) ≤ n − 1. In this paper, graphs with sdiam4( G) = 3, 4, n − 1 are characterized, respectively. [ABSTRACT FROM AUTHOR]

We show that the universal associative envelope of the simple anti-Jordan triple system of all ( is even, ) matrices over an algebraically closed field of characteristic 0 is finite-dimensional. The monomial basis and the center of the universal envelope are determined. The explicit decomposition of the universal envelope into matrix algebras is given. The classification of finite-dimensional irreducible representations of an anti-Jordan triple system is obtained. The semi-simplicity of the universal envelope is shown. We also show that the universal associative envelope of the simple polarized anti-Jordan triple system of matrices is infinite-dimensional. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of free objects in the variety of Steiner loops and of the combinatorial structures behind them, focusing on their automorphism groups. We prove that all automorphisms are tame and the automorphism group is not finitely generated if the loop is more than 3-generated. For the free Steiner loop with three generators we describe the generator elements of the automorphism group and some relations between them. [ABSTRACT FROM AUTHOR]

Let G be a graph and W a subset of V(G). A subtree with the minimum number of edges that contains all vertices of W is a Steiner tree for W. The number of edges of such a tree is the Steiner distance of W and union of all vertices belonging to Steiner trees for W form a Steiner interval. We describe both of these for the lexicographic product of graphs. We also give a complete answer for the following invariants with respect to the Steiner convexity: the Steiner number, the rank, the hull number, and the Carathéodory number, and a partial answer for the Radon number. [ABSTRACT FROM AUTHOR]

The frequency spectrums are inefficiently utilized and cognitive radio has been proposed for full utilization of these spectrums. The central idea of cognitive radio is to allow the secondary user to use the spectrum concurrently with the primary user with the compulsion of minimum interference. However, designing a model with minimum interference is a challenging task. In this paper, a transmission model based on cyclic generalized polynomial codes discussed in [2] and [15], is proposed for the improvement in utilization of spectrum. The proposed model assures a non interference data transmission of the primary and secondary users. Furthermore, analytical results are presented to show that the proposed model utilizes spectrum more efficiently as compared to traditional models. [ABSTRACT FROM AUTHOR]

Given a point set K of terminals in the plane, the octilinear Steiner tree problem is to find a shortest tree that interconnects all terminals and edges run either in horizontal, vertical, or ±45° diagonal direction. This problem is fundamental for the novel octilinear routing paradigm in VLSI design, the so-called X-architecture. As the related rectilinear and the Euclidian Steiner tree problem are well-known to be NP-hard, the same was widely believed for the octilinear Steiner tree problem but left open for quite some time. In this paper, we prove the NP-completeness of the decision version of the octilinear Steiner tree problem. We also show how to reduce the octilinear Steiner tree problem to the Steiner tree problem in graphs of polynomial size with the following approximation guarantee. We construct a planar graph of size $O(\frac{n^2}{\varepsilon^2})$ which contains a (1 + ε)–approximation of a minimum octilinear Steiner tree for every ε > 0 and n = |K|. Hence, we can apply any α–approximation algorithm for the Steiner tree problem in graphs (for planar graphs, Borradaile et al. very recently presented a polynomial time approximation scheme) and achieve an (α + ε)–approximation bound for the octilinear Steiner tree problem. This approximation guarantee also holds for the more difficult case where the Steiner tree has to avoid blockages (obstacles bounded by octilinear polygons). [ABSTRACT FROM AUTHOR]

The Euclidean centre (centre of the smallest enclosing sphere) of a set of points P in two or more dimensions is unstable; small perturbations at only a few points of P can result in an arbitrarily large relative change in the position of the Euclidean centre. Any centre function more stable than the Euclidean centre is eccentric; that is, its associated radius exceeds the radius of the smallest enclosing circle for some point sets P. Motivated by applications in mobile facility location (in which clients move continuously with some maximum velocity) we seek alternative notions of centrality that are stable while maintaining low eccentricity. In general there is a trade-off; centre functions with lower eccentricity are less stable. In an attempt to balance the conflicting goals of closeness of approximation and stability, we apply the Steiner centre, traditionally defined for convex polytopes, as a centre function of a set of points in the plane. Although previously defined, the notion of a Steiner centre had not been analyzed in terms of its approximation of the Euclidean centre. Exploiting the equivalence of the two definitions of the Steiner centre established by Shephard,27 we prove the stability of the Steiner centre is π/4 and show that the associated radius is at most 1.1153 times the Euclidean radius of any point set P. It follows that a mobile facility located at the Steiner centre of the positions of a set of mobile clients remains close to the Euclidean centre of the clients yet never moves with relative velocity that exceeds 4/π. [ABSTRACT FROM AUTHOR]

We introduce a new algorithm to convert triangular meshes of polygonal regions, with or without holes, into strictly convex quadrilateral meshes of small bounded size. Our algorithm includes all vertices of the triangular mesh in the quadrilateral mesh, but may add extra vertices (called Steiner points). We show that if the input triangular mesh has t triangles, our algorithm produces a mesh with at most $\lfloor\frac{3t}{2}\rfloor+2$ quadrilaterals by adding at most t+2 Steiner points, one of which may be placed outside the triangular mesh domain. We also describe an extension of our algorithm to convert constrained triangular meshes into constrained quadrilateral ones. We show that if the input constrained triangular mesh has t triangles and its dual graph has h connected components, the resulting constrained quadrilateral mesh has at most $\lfloor\frac{3t}{2}\rfloor+4h$ quadrilaterals and at most t+3h Steiner points, one of which may be placed outside the triangular mesh domain. Examples of meshes generated by our algorithm, and an evaluation of the quality of these meshes with respect to a quadrilateral shape quality criterion are presented as well. [ABSTRACT FROM AUTHOR]

For a given set of points in the Euclidean plane, a minimum network (a Steiner minimal tree) can be constructed using a geometric method, called Melzak's construction. The core of the Melzak construction is to replace a pair of terminals adjacent to the same Steiner point with a new terminal. In this paper we prove that the Melzak construction can be generalized to constructing Steiner minimal trees for circles so that either the given points (terminals) are constrained on the circles or the terminal edges are tangent to the circles. Then we show that the generalized Melzak construction can be used to find minimum networks separating and surrounding circular objects or to find minimum networks connecting convex and smoothly bounded objects and avoiding convex and smoothly bounded obstacles. [ABSTRACT FROM AUTHOR]

No abstract received. [ABSTRACT FROM AUTHOR]