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The paper gives an overview of a number of mathematical models and problems on planning anti-terrorist and special actions. These are problems of monitoring a territory, protecting critical infrastructure (described by optimization models), interdicting transport and information networks. It is shown that many territory control problems can be reduced to well-known optimization problems on graphs, shortest paths search, and minimal covering on graphs. The problems of protecting critical infrastructure and interdicting networks are reduced to stochastic minimax game problems. [ABSTRACT FROM AUTHOR]

In this paper we introduce the 3-way intersection problem for G-designs, and we consider this problem for kite systems. Let bv=v(v-1)/8 and I3(v)={0,1,…,bv}\{bv-1}. Let J3(v)={s: there exist three kite systems of order v intersecting in s blocks}. We show that for any positive integer v≡0,1(mod8) and v≥8, J3(v)=I3(v). [ABSTRACT FROM AUTHOR]

The fundamental group of the complement of a hyperplane arrangement in a complex vector space is an important topological invariant. The third rank of successive quotients in the lower central series of the fundamental group was called Falk invariant of the arrangement since Falk gave the first combinatorial formula and asked to give a combinatorial interpretation. In this article we prove that the Falk invariant of an arrangement associated with signed graph G without loops is double of the sum of k3, k4 and k3±, where kl is the number of subgraphs of G that are switching equivalent to the cliques with l vertices, k3± is that of subgraphs which have 3 vertices and each two vertices are connected by double edges with different signs. This formula modifies the one given by Schenck and Suciu, and answers partially Falk’s question in the case of signed graphic arrangements. [ABSTRACT FROM AUTHOR]

We consider latin square graphs Γ=LSG(H) of the Cayley table of a given finite group H. We characterize all pairs (Γ,G), where G is a subgroup of autoparatopisms of the Cayley table of H such that G acts arc-transitively on Γ and all nontrivial G-normal quotient graphs of Γ are complete. We show that H must be elementary abelian and determine the number k of complete normal quotients. This yields new infinite families of diameter two arc-transitive graphs with k=1 or k=2. [ABSTRACT FROM AUTHOR]

The independence polynomialI(G, x) of a finite graph G is the generating function for the sequence of the number of independent sets of each cardinality. We investigate whether, given a fixed number of vertices and edges, there exists optimally-least (optimally-greatest) graphs, that are least (respectively, greatest) for all non-negative x. Moreover, we broaden our scope to k-independence polynomials, which are generating functions for the k-clique-free subsets of vertices. For k≥3, the results can be quite different from the k=2 (i.e. independence) case. [ABSTRACT FROM AUTHOR]

The eigenvalues of a graph are algebraic integers in some algebraic extension of the rationals. We investigate the algebraic degree of these eigenvalues with respect to graph-theoretical properties. We obtain quantitative results showing that a graph with large diameter must have some eigenvalues of large algebraic degree. [ABSTRACT FROM AUTHOR]

An antimagic labelling of a graph G with m edges is a bijection f:E(G)→{1,…,m} such that for any two distinct vertices u, v we have ∑e∈E(v)f(e)≠∑e∈E(u)f(e), where E(v) denotes the set of edges incident v. The well-known Antimagic Labelling Conjecture formulated in 1994 by Hartsfield and Ringel states that any connected graph different from K2 admits an antimagic labelling. A weaker local version which we call the Local Antimagic Labelling Conjecture says that if G is a graph distinct from K2, then there exists a bijection f:E(G)→{1,…,|E(G)|} such that for any two neighbours u, v we have ∑e∈E(v)f(e)≠∑e∈E(u)f(e). This paper proves the following more general list version of the local antimagic labelling conjecture: Let G be a connected graph with m edges which is not a star. For any list L of m distinct real numbers, there is a bijection f:E(G)→L such that for any pair of neighbours u, v we have that ∑e∈E(v)f(e)≠∑e∈E(u)f(e). [ABSTRACT FROM AUTHOR]

A dynamic coloring of the vertices of a graph G starts with an initial subset S of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set S is called a forcing set (zero forcing set) of G if, by iteratively applying the forcing process, every vertex in G becomes colored. If the initial set S has the added property that it induces a subgraph of G without isolated vertices, then S is called a total forcing set in G. The total forcing number of G, denoted Ft(G), is the minimum cardinality of a total forcing set in G. We prove that if G is a connected, claw-free, cubic graph of order n≥6, then Ft(G)≤12n, where a claw-free graph is a graph that does not contain K1,3 as an induced subgraph. The graphs achieving equality in these bounds are characterized. [ABSTRACT FROM AUTHOR]

We consider the well-known problem of maximum of the functionalInγ=rγB0.0∏k=1nrBkak,where B0, …, Bn are pairwise disjoint domains in ℂ¯, a0 = 0, |ak| = 1, k=1,n¯, are different points of the circle, γ ∈ (0, n], and r(B, a) is the inner radius of the domain B⊂ℂ¯ relative to the point a. In the case of simply connected domains for n=2, 3, and 4, we have obtained the solution of this problem for the maximum interval of values of the parameter γ. [ABSTRACT FROM AUTHOR]

We consider one of the classical problems of the geometric theory of functions of a complex variable on a maximum of the functionalrB0.0rB∞∞γ∏k=1nrBkak,where n ∈ ℕ, n ≥ 2, γ ∈ ℝ+, An=akk=1n is a system of points such that |ak| = 1, a0 = 0, B0, B∞, Bkk=1n is a system of pairwise nonoverlapping domains, ak∈Bk⊂ℂ¯, k=0,n¯, ∞∈B∞⊂ℂ¯, r(B, a) is the inner radius of the domain B⊂ℂ¯ with respect to the point a ∈ B. We have analyzed this problem under some weaker restrictions on pairwise nonoverlapping domains. [ABSTRACT FROM AUTHOR]

We investigate the complex spectra XA(β)=∑j=0najβj:n∈N,aj∈Awhere β is a quadratic or cubic Pisot-cyclotomic number and the alphabet A is given by 0 along with a finite collection of roots of unity. Such spectra are discrete aperiodic structures with crystallographically forbidden symmetries. We discuss in general terms under which conditions they possess the Delone property required for point sets modeling quasicrystals. We study the corresponding Voronoi tilings and we relate these structures to quasilattices arising from the cut-and-project method. [ABSTRACT FROM AUTHOR]

We consider the half-wave maps equation ∂tS→=S→∧|∇|S→,where S→=S→(t,x) takes values on the two-dimensional unit sphere S2 and x∈R (real line case) or x∈T (periodic case). This an energy-critical Hamiltonian evolution equation recently introduced in Lenzmann and Schikorra (2017, arXiv:1702.05995v2), Zhou and Stone (Phys Lett A 379:2817-2825, 2015) which formally arises as an effective evolution equation in the classical and continuum limit of Haldane-Shastry quantum spin chains. We prove that the half-wave maps equation admits a Lax pair and we discuss some analytic consequences of this finding. As a variant of our arguments, we also obtain a Lax pair for the half-wave maps equation with target H2 (hyperbolic plane). [ABSTRACT FROM AUTHOR]

Given a sequence S=(s1,…,sm)∈[0,1]m, a block B of S is a subsequence B=(si,si+1,…,sj). The size b of a block B is the sum of its elements. It is proved in [1] that for each positive integer n, there is a partition of S into n blocks B1, …,Bn with |bi-bj|≤1 for every i, j. In this paper, we consider a generalization of the problem in higher dimensions. [ABSTRACT FROM AUTHOR]

A sufficient condition for the hyperbolicity of a group presented in terms of generators and defining relations is considered. The condition is formulated in terms of the negativity of a discrete analog of curvature for the Lyndon-van Kampen diagrams over a presentation of a group and is a generalization of the small cancellation condition. [ABSTRACT FROM AUTHOR]

A b-coloring of the vertices of a graph is a proper coloring where each color class contains a vertex which is adjacent to each other color class. The b-chromatic number of G is the maximum integer $$b(G)$$ for which G has a b-coloring with $$b(G)$$ colors. A graph G is b-continuous if G has a b-coloring with k colors, for every integer k in the interval $$[\chi (G),b(G)]$$ . It is known that not all graphs are b-continuous, and that it is NP-complete to decide whether a given graph G is b-continuous even if $$\chi (G)$$ and $$b(G)$$ are known. Also, there are many results that show that finding b-colorings of graphs with large girth is an easier task. For instance, finding $$b(G)$$ can be done in polynomial time when G has girth at least 7; also, regular graphs with girth at least 8 are b-continuous. In this article, we show that if G has girth at least 10, then G is b-continuous. [ABSTRACT FROM AUTHOR]

This paper generalizes the results in Aswal et al. (Econ Theory 22:45-62, ) on dictatorial domains. This is done in two ways. In the first, the notion of connections between pairs of alternatives in Aswal et al. () is weakened to weak connectedness. This notion requires the specification of four preference orderings for every alternative pair. Domains that are linked in the sense of Aswal et al. () with weak connectedness replacing connectedness, are shown to be dictatorial. In the second, the notion of connections for alternative pairs is strengthened relative to its counterpart in Aswal et al. (). However, a domain is shown to be dictatorial if the induced graph is merely connected rather than linked. This result generalizes the result in Sato (Rev Econ Design 14:331-342, ) on circular domains. [ABSTRACT FROM AUTHOR]

We prove that, if f : Rn → R satisfies Fréchet's functional equation Δm+1h f(x) = 0 for all x = (x1, ... , xn), h = (h1, ... , hn) ∊ Rn, and f(x1, ... , xn) is not an ordinary algebraic polynomial in the variables x1, ... , xn, then f is unbounded on all non-empty open set U ⊆ Rn. Furthermore, the set G(f)Rn+1 contains an unbounded open set. [ABSTRACT FROM AUTHOR]

Tutte conjectured that every 4-edge-connected graph admits a nowhere-zero 3-flow. In this paper, we show that this conjecture is true for Cayley graph on generalized dihedral groups and generalized quaternion groups, which generalizes the result of F. Yang and X. Li [Inform. Process. Lett., 2011, 111: 416-419]. We also generalizes an early result of M. Nánásiová and M. Škoviera [J. Algebraic Combin., 2009, 30: 103-110]. [ABSTRACT FROM AUTHOR]

We discuss stochastic functional partial differential equations and neutral partial differential equations of retarded type driven by fractional Brownian motion with Hurst parameter H > 1/2. Using the Girsanov transformation argument, we establish the quadratic transportation inequalities for the law of the mild solution of those equations driven by fractional Brownian motion under the L metric and the uniform metric. [ABSTRACT FROM AUTHOR]

The bondage number b( G) of a graph G is the cardinality of a minimum set of edges whose removal from G results in a graph with a domination number greater than that of G. In this paper, we obtain the exact value of the bondage number of the strong product of two paths. That is, for any two positive integers m ⩾ 2 and n ⩾ 2, b( P ⊠ P) = 7 − r( m) − r( n) if ( r( m), r( n)) = (1, 1) or (3, 3), 6 − r( m) − r( n) otherwise, where r( t) is a function of positive integer t, defined as r( t) = 1 if t ≡ 1 (mod 3), r( t) = 2 if t ≡ 2 (mod 3), and r( t) = 3 if t ≡ 0 (mod 3). [ABSTRACT FROM AUTHOR]

We give the condition of isomorphisms between tilting graphs and cluster-tilting graphs of hereditary algebras. As a conclusion, it is proved that a graph is a skeleton graph of Stasheff polytope if and only if it is both the tilting graph of a hereditary algebra and also the cluster-tilting graph of another hereditary algebra. At last, when comparing such uniformity, the geometric realizations of simplicial complexes associated with tilting modules and cluster-tilting objects are discussed respectively. [ABSTRACT FROM AUTHOR]

We study biplane graphs drawn on a finite point set $$S$$ in the plane in general position. This is the family of geometric graphs whose vertex set is $$S$$ and which can be decomposed into two plane graphs. We show that every sufficiently large point set admits a 5-connected biplane graph and that there are arbitrarily large point sets that do not admit any 6-connected biplane graph. Furthermore, we show that every plane graph (other than a wheel or a fan) can be augmented into a 4-connected biplane graph. However, there are arbitrarily large plane graphs that cannot be augmented to a 5-connected biplane graph by adding pairwise noncrossing edges. [ABSTRACT FROM AUTHOR]

Given a set of points in the plane each colored either red or blue, we find non-self-intersecting geometric spanning cycles of the red points and of the blue points such that each edge of the red spanning cycle is crossed at most three times by the blue spanning cycle and vice-versa. [ABSTRACT FROM AUTHOR]

The eigenvalue problem for the Laplacian with Dirichlet boundary conditions on a graph is considered. The main result is an estimate of the first (minimal) eigenvalue. The proof is based on the Schwartz symmetrization of a function on a graph and on its properties. [ABSTRACT FROM AUTHOR]

A plane graph G is edge-face k-colorable if the elements of $${E(G) \cup F(G)}$$ can be colored with k colors so that any two adjacent or incident elements receive different colors. Sanders and Zhao conjectured that every plane graph with maximum degree Δ is edge-face (Δ + 2)-colorable and left the cases $${\Delta \in \{4, 5, 6\}}$$ unsolved. In this paper, we settle the case Δ = 6. More precisely, we prove that every plane graph with maximum degree 6 is edge-face 8-colorable. [ABSTRACT FROM AUTHOR]

We developed a code for imaging the surfaces of spotted stars by a set of circular spots with a uniform temperature distribution. The flux from the spotted surface is computed by partitioning the spots into elementary areas. The code takes into account the passing of spots behind the visible stellar limb, limb darkening, and overlapping of spots. Modeling of light curves includes the use of recent results of the theory of stellar atmospheres needed to take into account the temperature dependence of flux intensity and limb darkening coefficients. The search for spot parameters is based on the analysis of several light curves obtained in different photometric bands. We test our technique by applying it to HII 1883. [ABSTRACT FROM AUTHOR]

Let G be a planar graph with maximum degree Δ. In this paper, we prove that if any 4-cycle is not adjacent to an i-cycle for any i ∈ {3, 4} in G, then the list edge chromatic number xl′ ( G) = Δ and the list total chromatic number xl″ ( G) = Δ+1. [ABSTRACT FROM AUTHOR]

This article presents a family of semidefinite programming bounds, obtained by Lagrangian duality, for 0-1 quadratic optimization problems with linear or quadratic constraints. These bounds have useful computational properties: they have a good ratio of tightness to computing time, they can be optimized by a quasi-Newton method, and their final tightness level is controlled by a real parameter. These properties are illustrated on three standard combinatorial optimization problems: unconstrained 0-1 quadratic optimization, heaviest $$k$$-subgraph, and graph bisection. [ABSTRACT FROM AUTHOR]

The base graph of a simple matroid M = ( E,B) is the graph G such that V ( G) = B and E( G) = { BB′: B,B′ ∈ B, | B / B′| = 1}, where the same notation is used for the vertices of G and the bases of M. It is proved that the base graph G of connected simple matroid M is Z-connected if | V ( G)| ⩽ 5. We also proved that if M is not a connected simple matroid, then the base graph G of M does not admit a nowhere-zero 3-flow if and only if | V ( G)| = 4. Furthermore, if for every connected component E ( i ⩽ 2) of M, the matroid base graph G of M = M| E has | V ( G)| ⩽ 5, then G is Z-connected which also implies that G admits nowhere-zero 3-flow immediately. [ABSTRACT FROM AUTHOR]

Let G be a finite group and π(G) = {p1, p2, ... , ps}. For p ∊ π(G), we put deg(p) : = ∣{q ∊ π(G)∣p ~ q in the prime graph of G}∣, which is called the degree of p. We also define D(G) := (deg(p1),deg(p2), ... ,deg(ps)), where p1 < p2 < ... < ps, which is called the degree pattern of G. We say G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups having the same order and degree pattern as G. In particular, a 1-fold OD-characterizable group is simply called an OD-characterizable group. In the present paper, we determine all finite simple groups whose first prime graph components are 1-regular and prove that all finite simple groups whose first prime graph components are r-regular except U4(2) are OD-characterizable, where 0 ≤ r ≤ 2. In particular, U4(2) is exactly 2-fold OD-characterizable, which improves an earlier obtained result. [ABSTRACT FROM AUTHOR]

In this paper, we establish a Marcinkiewicz-Zygmund type strong law for sequences of blockwise and pairwise negative quadrant m-dependent random variables. The sharpness of the results is illustrated by an example. [ABSTRACT FROM AUTHOR]

Let D be a finite and simple digraph with vertex set V(D), and let f : V(D)→ {-1,1} be a two-valued function. If k ≥ 1 is an integer and Σα∊N-[v] f (x) ≤ k for each v ∊V(D), where N-[v] consists of v and all vertices of D from which arcs go into v, then f is a signed k-dominating function on D. A set {f1, f2, ..., fd} of distinct signed k-dominating functions of D with the property that Σdi=1 fi(v) ≤ 1 for each v ∊ V(D), is called a signed k-dominating family (of functions) of D. The maximum number of functions in a signed k-dominating family of D is the signed k-domatic number of D, denoted by dkS(D). In this note we initiate the study of the signed k-domatic numbers of digraphs and present some sharp upper bounds for this parameter. [ABSTRACT FROM AUTHOR]

Let TX be the full transformation semigroup on a set X and E be a non-trivial equivalence on X. The set TE (X) = { f ∊ TX : ∀(x,y) ∊ E; ( f (x), f (y)) ∊ E} is a subsemigroup of TX . For a finite totally ordered set X and a convex equivalence E on X, the set of all the orientation-preserving transformations in TE(X) forms a subsemigroup of TE(X) denoted by OPE(X). In this paper, under the hypothesis that the totally ordered set X is of cardinality mn (m,n ≥ 2) and the equivalence E has m classes such that each E-class contains n consecutive points, we calculate the cardinality of the semigroup OPE(X), and that of its idempotents. [ABSTRACT FROM AUTHOR]

Some new relations have been established between Wiener indices, stability numbers and clique numbers for several classes of composite graphs that arise via graph products. For three of considered operations we show that they make a multiplicative pair with the clique number. [ABSTRACT FROM AUTHOR]

Let (R, m) be a commutative Noetherian local ring, a an ideal of R, and M a finitely generated R-module. We show that for a non-negative integer t the following cases are equivalent: (a) The formal local cohomology modules lim← n Him(M/anM) are Artinian for all i < t; (b) a ⊆ Rad(Ann(lim←n Him (M/anM))) for all i < t. If one of the above cases holds, then lim←n Htm(M=anM)/alim←n Htm (M/anM) is Artinian. Also, there are some results concerning finiteness properties of formal local cohomology modules. [ABSTRACT FROM AUTHOR]

For a locally compact group G, L1(G) is its group algebra and L∞(G) is the dual of L1(G). Crombez and Govaerts introduced the notion of a uniformly measurable function in L∞(G) and proved that such a function induces a completely continuous operator. The aim of this paper is to go further and generalize the above results to foundation semigroup algebras. We study completely continuous linear maps on semigroup algebras which commute with translations. [ABSTRACT FROM AUTHOR]

For a positive integer k, a total {k}-dominating function of a graph G without isolated vertices is a function f from the vertex set V(G) to the set {0,1,2, . . . ,k} such that for any vertex v ∊ V(G), the condition Σu∊N(v) f (u) ≥ k is fulfilled, where N(v) is the open neighborhood of v. The weight of a total {k}-dominating function f is the value ω(f) = Σv∊V f (v). The total {k}-domination number, denoted by γt{k} (G), is the minimum weight of a total {k}-dominating function on G. A set {f1, f2, . . . , fd} of total {k}-dominating functions on G with the property that Σdi =1 fi(v) ≤ k for each v ∊ V(G), is called a total {k}-dominating family (of functions) on G. The maximum number of functions in a total {k}-dominating family on G is the total {k}-domatic number of G, denoted by dt{k} (G). Note that dt{1} (G) is the classic total domatic number dt(G). In this paper, we present bounds for the total {k}-domination number and total {k}-domatic number. In addition, we determine the total {k}-domatic number of cylinders and we give a Nordhaus-Gaddum type result. [ABSTRACT FROM AUTHOR]

The second Zagreb index of a graph G is an adjacency-based topological index, which is defined as Σuv∊E(G)(d(u)d(v)), where uv is an edge of G, d(u) is the degree of vertex u in G. In this paper, we consider the second Zagreb index for bipartite graphs. Firstly, we present a new definition of ordered bipartite graphs, and then give a necessary condition for a bipartite graph to attain the maximal value of the second Zagreb index. We also present an algorithm for transforming a bipartite graph to an ordered bipartite graph, which can be done in O(n2+n21) time for a bipartite graph B with a partition ∣X∣ = n1 and ∣Y∣ = n2. [ABSTRACT FROM AUTHOR]

A graph G satisfies the Ore-condition if d( x)+ d( y) ≥ | V( G)| for any xy ∉ E( G). Luo et al. [ European J. Combin., 2008 characterized the simple Z-connected graphs satisfying the Ore-condition. In this paper, we characterize the simple Z-connected graphs G satisfying d( x) + d( y) ≥ | V( G)| − 1 for any xy ∈ E( G), which improves the results of Luo et al. [ABSTRACT FROM AUTHOR]

Let ℋ be a hypergraph with maximal vertex degree Δ such that each its hyperedge has at least δ vertices. Let $ k = \left\lceil {\frac{{2\Delta }}{\delta }} \right\rceil $. We prove that ℋ admits a proper vertex coloring with k + 1 colors (i.e., such that any hyperedge contains at least two vertices of different colors). For k ≥ 3 and δ ≥ 3 we prove that ℋ admits a proper vertex coloring with k colors.For a graph G set $ k = \left\lceil {\frac{{2\Delta (G)}}{{\delta (G)}}} \right\rceil $. As a corollary, we prove that there exists a dynamic coloring of the graph G with k + 1 colors in general and with k colors if δ( G) ≥ 3 and k ≥ 3. Bibliography: 16 titles. [ABSTRACT FROM AUTHOR]

We analyze a SD-equation on a graph from the viewpoint of variational laws and generalize the mathematical modeling on space grids to the case of an irregular object. We take into account the possible appearance of the delta-function phenomenon even at the grid nodes. [ABSTRACT FROM AUTHOR]

A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. In this paper, it is shown that each 1-planar graph with minimum degree 7 contains a copy of K∨( K∪ K) with all vertices of degree at most 12. In addition, we also prove the existence of a graph K∨( K∪ K) with relatively small degree vertices in 1-planar graphs with minimum degree at least 6. [ABSTRACT FROM AUTHOR]

Consider the NP-hard problem of, given a simple graph G, to find a series-parallel subgraph of G with the maximum number of edges. The algorithm that, given a connected graph G, outputs a spanning tree of G, is a $\frac{1}{2}$-approximation. Indeed, if n is the number of vertices in G, any spanning tree in G has n−1 edges and any series-parallel graph on n vertices has at most 2 n−3 edges. We present a $\frac{7}{12}$-approximation for this problem and results showing the limits of our approach. [ABSTRACT FROM AUTHOR]

In this series of papers we show that there are exactly ten subfactors, other than A subfactors, of index between 4 and 5. Previously this classification was known up to index $${3+\sqrt{3}}$$. In the first paper we give an analogue of Haagerup's initial classification of subfactors of index less than $${3+\sqrt{3}}$$, showing that any subfactor of index less than 5 must appear in one of a large list of families. These families will be considered separately in the three subsequent papers in this series. [ABSTRACT FROM AUTHOR]

We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in $${{\mathbb R}^{2n}}$$ , we show that the parabolic Eq. 1.1 has a longtime solution which is smooth for all positive time and satisfies uniform estimates away from time t = 0. In particular, under the mean curvature flow (1.2) the graph immediately becomes smooth and the solution exists for all time such that the second fundamental form decays uniformly to 0 on the graph as t → ∞. Our assumption on the Lipschitz norm is equivalent to the underlying Lagrangian potential u being uniformly convex with its Hessian bounded in L. As an application of this result we provide conditions under which an entire Lipschitz Lagrangian graph converges after rescaling to a self-expanding solution to the mean curvature flow. [ABSTRACT FROM AUTHOR]

In the present paper, we identify the first two minimal with respect to set-inclusion hereditary classes of graphs of unbounded clique-width: Bipartite permutation graphs and unit interval graphs. [ABSTRACT FROM AUTHOR]

We define 2-decompositions of ribbon graphs, which generalize 2-sums and tensor products of graphs. We give formulae for the Bollobás-Riordan polynomial of such a 2-decomposition, and derive the classical Brylawski formula for the Tutte polynomial of a tensor product as a (very) special case. This study was initially motivated from knot theory, and we include an application of our formulae to mutation in knot diagrams. [ABSTRACT FROM AUTHOR]

We introduce a new class of countably infinite random geometric graphs, whose vertices V are points in a metric space, and vertices are adjacent independently with probability $${p \in (0, 1)}$$ if the metric distance between the vertices is below a given threshold. For certain choices of V as a countable dense set in $${\mathbb{R}^n}$$ equipped with the metric derived from the L-norm, it is shown that with probability 1 such infinite random geometric graphs have a unique isomorphism type. The isomorphism type, which we call GR, is characterized by a geometric analogue of the existentially closed adjacency property, and we give a deterministic construction of GR. In contrast, we show that infinite random geometric graphs in $${\mathbb{R}^{2}}$$ with the Euclidean metric are not necessarily isomorphic. [ABSTRACT FROM AUTHOR]

In this paper we study non-planar drawings of graphs, and study trade-offs between the crossing resolution (i.e., the minimum angle formed by two crossing segments), the curve complexity (i.e., maximum number of bends per edge), the total number of bends, and the area. [ABSTRACT FROM AUTHOR]

A set Q⊆ V is a hub set of a graph G=( V, E) if, for every pair of vertices u, v∈ V∖ Q, there exists a path from u to v such that all intermediate vertices are in Q. The hub number of G is the minimum size of a hub set in G. This paper derives the hub numbers of Sierpiński-like graphs including: Sierpiński graphs, extended Sierpiński graphs, and Sierpiński gasket graphs. Meanwhile, the corresponding minimum hub sets are also obtained. [ABSTRACT FROM AUTHOR]