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Graph operations play an important role to constructing complex network structures from simple graphs, and these complex networks play vital roles in different fields such as computer science, chemistry, and social sciences. Computation of topological indices of these complex network structures via graph operation is an important task. In this study, we defined two new variants of graph products, namely, corona join and subdivision vertex join products and investigated exact expressions of the first and second Zagreb indices and first reformulated Zagreb index for these new products.

Let G be a simple, connected graph and let A(G) be the adjacency matrix of G. If D(G) is the diagonal matrix of the vertex degrees of G, then for every real $$\alpha \in [0,1]$$ , the matrix $$A_{\alpha }(G)$$ is defined as $$A_{\alpha }(G) = \alpha D(G) + (1- \alpha ) A(G).$$ The eigenvalues of the matrix $$A_{\alpha }(G)$$ form the $$A_{\alpha }$$ -spectrum of G. Let $$G_1 {\dot{\vee }} G_2$$ , $$G_1 {\underline{\vee }} G_2$$ , $$G_1 \langle \text {v} \rangle G_2$$ and $$G_1 \langle \text {e} \rangle G_2$$ denote the subdivision-vertex join, subdivision-edge join, R-vertex join and R-edge join of two graphs $$G_1$$ and $$G_2$$ , respectively. In this paper, we compute the $$A_{\alpha }$$ -spectra of $$G_1 {\dot{\vee }} G_2$$ , $$G_1 {\underline{\vee }} G_2$$ , $$G_1 \langle \text {v} \rangle G_2$$ and $$G_1 \langle \text {e} \rangle G_2$$ for a regular graph $$G_1$$ and an arbitrary graph $$G_2$$ in terms of their $$A_{\alpha }$$ -eigenvalues. As an application of these results, we construct infinitely many pairs of $$A_{\alpha }$$ -cospectral graphs.

Hypergraph is a generalization of graph in which an edge can join any number of vertices. Hypergraph is used for combinatorial structures which generalize graphs. In this research work, the notion of hypergraphical metric spaces is introduced, which generalizes many existing spaces. Some fixed point theorems are studied in the corresponding spaces. To show the authenticity of the established work, nontrivial examples and applications are also provided.

Let G be an edge-colored connected graph. A path P in G is called $$\ell $$ -rainbow if no two edges of the same color have fewer than $$\ell $$ edges between them on P. The graph G is called $$(k,\ell )$$ -rainbow connected if every pair of distinct vertices of G are connected by k pairwise internally vertex-disjoint $$\ell $$ -rainbow paths in G. For a k-connected graph G, the minimum number of colors needed to make G $$(k,\ell )$$ -rainbow connected is called the $$(k,\ell )$$ -rainbow connection number of G and denoted by $$rc_{k,\ell }(G)$$ . In this paper, we prove that $$rc_{1,2}(G)\le 5$$ for any 2-connected graph G and provide a counter example which overturns a conjecture posed by Chartrand et al. in 2016. Considering graph operations, we study the (1, 2)-rainbow connection numbers for the join, the Cartesian, strong and lexicographic products of graphs, giving the sharp upper bounds for nearly all graph products. In addition, we find some basic properties of the $$(k,\ell )$$ -rainbow connection number and determine the values of $$rc_{1,\ell }(G)$$ where G is a cube or a permutation graph of a nontrivial traceable graph.

Given a finite group G, the permutability graph of non-normal subgroups $$\Gamma _{N}(G)$$ is given by all proper non-normal subgroups of G as vertex set and two vertices H and K are joined if $$HK = KH$$ . Rajkumar and Devi generalized $$\Gamma _{N}(G)$$ to the permutability graph of subgroups $$\Gamma (G)$$ , extending the vertex set to all proper subgroups of G (removing the assumption of being non-normal) and keeping the same criterion to join two vertices. We consider a natural counterpart for $$\Gamma (G)$$ and $$\Gamma _{N}(G)$$ , that is, the subgroups lattice $$\mathrm {L}(G)$$ of G, and introduce the non-permutability graph of subgroups $$\Gamma _{\mathrm {L}(G)}$$ ; its vertices are now given by the set $$\mathrm {L}(G)-{\mathfrak {C}}_{\mathrm {L}(G)}(\mathrm {L}(G))$$ , where $${\mathfrak {C}}_{\mathrm {L}(G)}(\mathrm {L}(G))$$ denotes the smallest sublattice of $$\mathrm {L}(G)$$ containing all permutable subgroups of G, and we join two vertices H, K of $$\Gamma _{\mathrm {L}(G)}$$ if and only if $$HK\ne KH$$ . We study classical invariants of $$\Gamma _{\mathrm {L}(G)}$$ , showing generalizations of previous results in the literature.

We prove that the symmetrized deleted join SymmDelJoin( $$\mathcal{K}$$ ) of a “balanced family” $$\mathcal{K}$$ = 〈Ki〉 =1 of collectively r-unavoidable subcomplexes of 2[m] is (m−r−1)-connected. As a consequence we obtain a Tverberg-Van Kampen-Flores type result which is more conceptual and more general than previously known results. Already the case r = 2 of this result seems to be new as an extension of the classical Van Kampen-Flores theorem. The main tool used in the paper is R. Forman’s discrete Morse theory.

We extend, in significant ways, the theory of truncated boolean representable simplicial complexes introduced in 2015. This theory, which includes all matroids, represents the largest class of finite simplicial complexes for which combinatorial geometry can be meaningfully applied.

For any weighted directed path-cycle graphs, a and b (referred to as structures), and any equal costs of operations (intermergings and duplication), we obtain an algorithm which, by successively applying these operations to a, outputs b if the first structure contains no paralogs (i.e., edges with a repeated name) and the second has no more than two paralogs for each edge. In finding the shortest sequence of operations to be applied to pass from a to b, the algorithm has a multiplicative error of at most 13/9 + ε, where ε is any strictly positive number, and its runtime is of the order of nO(ε−2.6), where n is the size of the input pair of graphs. In the case of no paralogs, equal sets of names in the structures, and equal operation costs, we have considered the following conditions on the transformation of a into b: all structures in them are from one cycle; all structures are from one path; all structures are from paths. For each of the conditions, we have obtained an exact (i.e., zero-error) quadratic time algorithm for finding the shortest transformation of a into b. For another list of operations (join and cut of a vertex, and deletion and insertion of an edge) over structures and for arbitrary costs of these operations, we have obtained an algorithm for the extension of structures specified at the leaves of a tree onto its interior vertices. The algorithm is exact if the tree is a star—in this case, structures in the leaves may even have unequal sets of names or paralogs. The runtime of the algorithm is of the order of nΧ + n2log(n), where n is the number of names in the leaves, and Χ is an easily computable characteristic of the structures in the leaves. In the general case, a cubic time algorithm finds a locally minimal solution.

A star coloring of a graph [Formula: see text] is a proper coloring of [Formula: see text] such that no path of length [Formula: see text] in [Formula: see text] is bicolored. In this paper, the star chromatic number of join of graphs is computed. Some sharp bounds for the star chromatic number of corona, lexicographic, deleted lexicographic and hierarchical product of graphs together with a conjecture on the star chromatic number of lexicographic product of graphs are also presented.

Let G be a simple graph of order n with vertex set V(G) and edge set E(G), and let k be an integer such that 1≤k≤n−1. The k-token graph G{k} of G is the graph whose vertices are the k-subsets of V(G), where two vertices A and B are adjacent in G{k} whenever their symmetric difference A▵B, defined as (A∖B)∪(B∖A), is a pair {a,b} of adjacent vertices in G. In this paper we study the Hamiltonicity of the k-token graphs of some join graphs. We provide an infinite family of graphs, containing Hamiltonian and non-Hamiltonian graphs, for which their k-token graphs are Hamiltonian. Our result provides, to our knowledge, the first family of non-Hamiltonian graphs for which it is proven the Hamiltonicity of their k-token graphs, for any 2&lt, k&lt, n−2.

The Weihrauch degrees and strong Weihrauch degrees are partially ordered structures representing degrees of unsolvability of various mathematical problems. Their study has been widely applied in computable analysis, complexity theory, and more recently, also in computable combinatorics. We answer an open question about the algebraic structure of the strong Weihrauch degrees, by exhibiting a join operation that turns these degrees into a lattice. Previously, the strong Weihrauch degrees were only known to form a lower semi-lattice. We then show that unlike the Weihrauch degrees, which are known to form a distributive lattice, the lattice of strong Weihrauch degrees is not distributive. Therefore, the two structures are not isomorphic.

The main goal of this study is to characterize new classes of multicone graphs which are determined by their spectra. One of important part of algebraic graph theory is devoted to spectral graph theory. Determining whether a graph is determined by its spectra or not is often an important and challenging problem. In [ 1 ] it have been shown that the join of a Cocktail-Party graph with an arbitrary complete graph is determined by both its adjacency spectra and its Laplacian spectra. In this work, we aim to generalize these facts. A multicone graph is defined to be the join of a clique and a regular graph. Let w, m and n be natural numbers. In this paper, it is proved that any connected graph cospectral to a multicone graph Kw ▽ mCP(n) is determined by its adjacency spectra as well as its Laplacian spectra, where $ CP(n) = {K_{\underbrace {2, \, .\, .\, ., \, \, 2}_{n\, times}}}$ is a Cocktail-Party graph. Moreover, we prove that any graph cospectral to one of these multicone graphs must be perfect. Finally, we pose two conjectures for further research.

We study large-scale systems operating under the JSQ$(d)$ policy in the presence of stringent task-server compatibility constraints. Consider a system with $N$ identical single-server queues and $M(N)$ task types, where each server is able to process only a small subset of possible task types. Each arriving task selects $d\geq 2$ random servers compatible to its type, and joins the shortest queue among them. The compatibility constraint is naturally captured by a fixed bipartite graph $G_N$ between the servers and the task types. When $G_N$ is complete bipartite, the meanfield approximation is proven to be accurate. However, such dense compatibility graphs are infeasible due to their overwhelming implementation cost and prohibitive storage capacity requirement at the servers. Our goal in this paper is to characterize the class of sparse compatibility graphs for which the meanfield approximation remains valid. To achieve this, first, we introduce a novel graph expansion-based notion, called proportional sparsity, and establish that systems with proportionally sparse compatibility graphs match the performance of a fully flexible system, asymptotically in the large-system limit. Furthermore, for any $c(N)$ satisfying $$\frac{Nc(N)}{M(N)\ln(N)}\to \infty\quad \text{and}\quad c(N)\to \infty,$$ as $N\to\infty$, we show that proportionally sparse random compatibility graphs can be designed, so that the degree of each server is at most $c(N)$. This reduces the server-degree almost by a factor $N/\ln(N)$, compared to the complete bipartite compatibility graph, while maintaining the same asymptotic performance. Extensive simulation experiments are conducted to corroborate the theoretical results., Comment: 51 pages, 5 figures. 08/23/2020: Minor errors fixed in the proof of Theorem 4.3

In this paper, first we consider some partitioned matrices having some special kind of blocks, and we obtain their eigenvalues and eigenvectors. From these, we deduce several results on the eigenvalues of partitioned matrices in the literature. Next we consider an extension of a generalized join of graphs introduced by Hedetniemi (On classes of graphs defined by special cutsets of lines. In: Many facets of graph theory. Proceedings of the conference held at Western MiGammagan University, Kalamazoo/Mi, 1968. Lecture notes in mathematics, vol 110, pp 171–189, 1969), we call it the $${\mathcal {M}}$$ -join of $${\mathcal {H}}_k$$ . The spectra of partitioned matrices allow to detect the adjacency, the Laplacian and the signless Laplacian spectrum of old and new type of join of graphs. Further, we have constructed several pairs of simultaneously A-cospectral, L-cospectral and Q-cospectral graphs.

The distance Laplacian matrix of a simple connected graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the distance matrix of [Formula: see text] and [Formula: see text] is the diagonal matrix whose main diagonal entries are the vertex transmissions in [Formula: see text] In this paper, we determine the distance Laplacian spectra of the graphs obtained by generalization of the join and lexicographic product of graphs (namely joined union). It is shown that the distance Laplacian spectra of these graphs not only depend on the distance Laplacian spectra of the participating graphs but also depend on the spectrum of another matrix of vertex-weighted Laplacian kind (analogous to the definition given by Chung and Langlands [A combinatorial Laplacian with vertex weights, J. Combin. Theory Ser. A 75 (1996) 316–327]).

Given a finite group G, we denote by Δ(G) the graph whose vertices are the proper subgroups of G and in which two vertices H and K are joined by an edge if and only if G = ⟨H, K⟩. We prove that if there exists a finite nilpotent group X with Δ(G) ≅ Δ(X), then G is supersoluble.

In 2011, Caro et al. introduced the monochromatic connection of graphs. An edge-coloring of a connected graph $G$ is called a monochromatically connecting (MC-coloring, for short) if there is a monochromatic path joining any two vertices. The monochromatic connection number $\operatorname{mc}(G)$ of a graph $G$ is the maximum integer $k$ such that there is a $k$-edge-coloring, which is an MC-coloring of $G$. Clearly, a monochromatic spanning tree can monochromatically connect any two vertices. So for a graph $G$ of order $n$ and size $m$, $\operatorname{mc}(G) \geq m-n+2$. Caro et al. proved that both triangle-free graphs and graphs of diameter at least three meet the lower bound. ¶ In this paper, we consider the monochromatic connectivity of graphs containing triangles which meet the lower bound too. Also, in order to study the graphs of diameter two, we present the formula for the monochromatic connectivity of join graphs. This will be helpful to solve the problem for graphs of diameter two.

We consider the Type 1 and Type 2 noncommutative Borsuk-Ulam conjectures of Baum, D$\k{a}$browski, and Hajac: there are no equivariant morphisms $A \to A \circledast_\delta H$ or $H \to A \circledast_\delta H$, respectively, when $H$ is a nontrivial compact quantum group acting freely on a unital $C^*$-algebra $A$. Here $A \circledast_\delta H$ denotes the equivariant noncommutative join of $A$ and $H$; this join procedure is a modification of the topological join that allows a free action of $H$ on $A$ to produce a free action of $H$ on $A \circledast_\delta H$. For the classical case $H = \mathcal{C}(G)$, $G$ a compact group, we present a reduction of the Type 1 conjecture and counterexamples to the Type 2 conjecture. We also present some examples and conditions under which the Type 2 conjecture does hold., Comment: 15 pages. Version 4 has an expanded commentary in section 3, including the equivalence of a topological conjecture with a certain noncommutative generalization. To appear in Proceedings of the American Mathematical Society

The class CR of completely regular semigroups considered as algebras with binary multiplication and unary operation of inversion forms a variety. Kernel, trace, local and core relations, denoted by K, T, L and C, respectively, are quite useful in studying the structure of the lattice L(CR) of subvarieties of CR. They are equivalence relations whose classes are intervals. Their ends are used for defining operators on L(CR). Starting with a few band varieties, we repeatedly apply operators induced by upper ends of classes of these relations and characterize corresponding classes up to certain variety low in the lattice L(CR). We consider only varieties whose origin are “central” band varieties, that is those in the middle column of the lattice L(B) of band varieties. Several diagrams represent the (semi)lattices studied.

Finite monoids that generate monoid varieties with uncountably many subvarieties seem rare, and surprisingly, no finite monoid is known to generate a monoid variety with countably infinitely many subvarieties. In the present article, it is shown that there are, nevertheless, many finite monoids that generate monoid varieties with continuum many subvarieties; these include any finite inherently non-finitely based monoid and any monoid for which $xyxy$ is an isoterm. It follows that the join of two Cross monoid varieties can have a continuum cardinality subvariety lattice that violates the ascending chain condition. Regarding monoid varieties with countably infinitely many subvarieties, the first example of a finite monoid that generates such a variety is exhibited. A complete description of the subvariety lattice of this variety is given. This lattice has width three and contains only finitely based varieties, all except two of which are Cross.

We present a proof for certain cases of the noncommutative Borsuk-Ulam conjectures proposed by Baum, D\k{a}browski, and Hajac. When a unital $C^*$-algebra $A$ admits a free action of $\mathbb{Z}/k\mathbb{Z}$, $k \geq 2$, there is no equivariant map from $A$ to the $C^*$-algebraic join of $A$ and the compact "quantum" group $C(\mathbb{Z}/k\mathbb{Z})$. This also resolves D\k{a}browski's conjecture on unreduced suspensions of $C^*$-algebras. Finally, we formulate a different type of noncommutative join than the previous authors, which leads to additional open problems for finite cyclic group actions., Comment: 13 pages. To appear in IUMJ. Version 4 restructures the results to address earlier work of Volovikov

We prove that among all flag 3-manifolds on $n$ vertices, the join of two circles with $\left\lceil{\frac{n}{2}}\right \rceil$ and $\left\lfloor{\frac{n}{2}}\right \rfloor$ vertices respectively is the unique maximizer of the face numbers. This solves the first case of a conjecture due to Lutz and Nevo. Further, we establish a sharp upper bound on the number of edges of flag 5-manifolds and characterize the cases of equality. We also show that the inequality part of the flag upper bound conjecture continues to hold for all flag 3-dimensional Eulerian complexes and find all maximizers of the face numbers in this class., Comment: 13 pages

Paolo Aluffi, inspired by an algebro-geometric problem, asked when the Kirchhoff polynomial of a graph is in the Jacobian ideal of the Kirchhoff polynomial of the same graph with one edge deleted. We give some results on which graph-edge pairs have this property. In particular we show that multiple edges can be reduced to double edges, we characterize which edges of wheel graphs satisfy the property, we consider a stronger condition which guarantees the property for any parallel join, and we find a class of series-parallel graphs with the property., 36 pages. Typo correction as suggested by referee

Let $G$ be a finite $p$-group and $k$ be a field of characteristic $p$. A topological space $X$ is called an $n$-Moore space if its reduced homology is nonzero only in dimension $n$. We call a $G$-CW-complex $X$ an $\underline{n}$-Moore $G$-space over $k$ if for every subgroup $H$ of $G$, the fixed point set $X^H$ is an $\underline{n}(H)$-Moore space with coefficients in $k$, where $\underline{n}(H)$ is a function of $H$. We show that if $X$ is a finite $\underline{n}$-Moore $G$-space, then the reduced homology module of $X$ is an endo-permutation $kG$-module generated by relative syzygies. A $kG$-module $M$ is an endo-permutation module if ${\rm End}_k (M) =M \otimes _{k} M^*$ is a permutation $kG$-module. We consider the Grothendieck group of finite Moore $G$-spaces $\mathcal{M}(G)$, with addition given by the join operation, and relate this group to the Dade group generated by relative syzygies., 22 pages

In this paper, we associate a partial g-hypergroupoid with a given g-hypergraph and analyze the properties of this hyperstructure. We prove that a g-hypergroupoid may be a commutative hypergroup without being a join space. Next, we define diagonal direct product of g-hypergroupoids. Further, we construct a sequence of g-hypergroupoids and investigate some relationships between it?s terms. Also, we study the quotient of a g-hypergroupoid by defining a regular relation. Finally, we describe fundamental relation of an Hv-semigroup as a g-hypergroupoid.

Let $G$ be a simple graph on the vertex set $[n]$ and $J_G$ be the corresponding binomial edge ideal. Let $G=v*H$ be the cone of $v$ on $H$. In this article, we compute all the Betti numbers of $J_G$ in terms of Betti number of $J_H$ and as a consequence, we get the Betti diagram of wheel graph. Also, we study Cohen-Macaulay defect of $S/J_G$ in terms of Cohen-Macaulay defect of $S_H/J_H$ and using this we construct a graph with Cohen-Macaulay defect $q$ for any $q\geq 1$. We obtain the depth of binomial edge ideal of join of graphs. Also, we prove that for any pair $(r,b)$ of positive integers with $1\leq b< r$, there exists a connected graph $G$ such that $reg(S/J_G)=r$ and the number of extremal Betti number of $S/J_G$ is $b$., 16 pages, Typos corrected, To appear in Mathematische Nachrichten

In his “Marriages of Incommensurables: Phi Related Ratios Joined with the Square Roots of Two and Three”, artist and geometer Mark A. Reynolds has found two ratios from the golden section family that generate relationships with the square roots of two and three. He includes the grids and procedures necessary for producing these ratios. For him, the significance of the constructions is that they join together ratios from two different groups of rectangles: the golden section family and the square root rectangle progression, two systems that are usually incompatible with each other. In these constructions and grids, the square root of the golden section and the golden section squared are related mathematically to the square roots of two and three, respectively, in ways that he believes have not been seen before. Reynolds calls this series of constructions, “marriages of incommensurables”, and the two he presents here are part of a larger group he has been working on for some time.

We present a method of combining coupled cell systems to get dynamics supporting robust simple heteroclinic networks given by the product of robust simple heteroclinic networks (cycles). We consider coupled cell networks, with no assumption on symmetry, and combine them via the join operation. Assuming that the dynamics of the component networks supports robust simple heteroclinic cycles or networks, we show that the join dynamics realizes a more complex heteroclinic network given by the product of those cycles or networks. Moreover, the equilibria in the product heteroclinic network correspond to partially synchronous states. Assuming no symmetry for the component coupled cell networks, one of the key points for the existence and robustness of the heteroclinic dynamics is the flow-invariant subspaces forced by the network structure – the synchrony subspaces. The other key point is that the (linear) stability of equilibria in the join dynamics is determined by the (linear) stability of equilibria ...

It is well known that a locale is subfit iff each of its open sublocales is a join of closed ones, and fit iff each of its closed sublocales is a meet of open ones. This formulation, however, exaggerates the parallelism between the behavior of fitness and subfitness. For it can be shown that a locale is fit iff each of its sublocales is a meet of closed ones, but it is not the case that a locale is subfit iff each of its sublocales is a join of closed ones. Thus we are led to take up the very natural question of which locales have the feature that every sublocale is a join of closed sublocales. In this note we show that these are precisely the subfit locales which are scattered in the pointfree sense of (13), and we add a variation for spatial frames.

Let G 1 and G 2 be disjoint copies of a graph G and g : V ( G 1 ) &rarr, V ( G 2 ) be a function. A functigraph F G consists of the vertex set V ( G 1 ) &cup, V ( G 2 ) and the edge set E ( G 1 ) &cup, E ( G 2 ) &cup, { u v : g ( u ) = v } . In this paper, we extend the study of distinguishing numbers of a graph to its functigraph. We discuss the behavior of distinguishing number in passing from G to F G and find its sharp lower and upper bounds. We also discuss the distinguishing number of functigraphs of complete graphs and join graphs.

The purpose of this paper is to study reducibility properties in Sasakian geometry. First we give the Sasaki version of the de Rham Decomposition Theorem; however, we need a mild technical assumption on the Sasaki automorphism group which includes the toric case. Next we introduce the concept of {\it cone reducible} and consider $S^3$ bundles over a smooth projective algebraic variety where we give a classification result concerning contact structures admitting the action of a 2-torus of Reeb type. In particular, we can classify all such Sasakian structures up to contact isotopy on $S^3$ bundles over a Riemann surface of genus greater than zero. Finally, we show that in the toric case an extremal Sasaki metric on a Sasaki join always splits., Comment: 58 pages, minor corrections made in latest version; a reference added and references updated; to appear in the Transactions of the American Mathematical Society

Let S(V) be a complex linear sphere of a finite group G. Let S(V)*n denote the n-fold join of S(V) with itself and let aut G(S(V)*) denote the space of G-equivariant self-homotopy equivalences of S(V)*n. We show that for any k ≥ 1 there exists M > 0 that depends only on V such that |πk autG(S(V)*n)|≤ M for all n ≫0.

Let G be a finite group and \(H\le G\). In this paper, we investigate the influence of the index of H in \(\langle H, H^{g}\rangle \) on the structure of G.

The Bernstein operator B n for a simplex in R d is naturally defined via the Bernstein basis obtained from the barycentric coordinates given by its vertices. Here we consider a generalisation of this basis and the Bernstein operator, which is obtained from generalised barycentric coordinates that are given for more general configurations of points in R d . We call the associated polynomials a Bernstein frame, as they span the polynomials of degree ? n , but may not be a basis. By using this redundant system we are able to give geometrically motivated proofs of some basic properties of the corresponding generalised Bernstein operator, such as the fact it is degree reducing and converges for all polynomials. We also consider the conditions for polynomials in this Bernstein form to join smoothly.

In this paper, we study a complete noncompact nonnegatively curved Alexandrov space $A$ with a soul $S$ of codimension two. We establish some structural results under additional regularity assumptions. As an application, we conclude that in this case Sharafutdinov retraction, $\pi:\ A\to S$, is a submetry., Comment: 30 pages

We prove a common generalization of the Ham Sandwich theorem and Alon's Necklace Splitting theorem. Our main results show the existence of fair distributions of $m$ measures in $R^d$ among $r$ thieves using roughly $mr/d$ convex pieces, even in the cases when $m$ is larger than the dimension. The main proof relies on a construction of a geometric realization of the topological join of two spaces of partitions of $R^d$ into convex parts, and the computation of the Fadell-Husseini ideal valued index of the resulting spaces., 14 pages, 9 figures

In this paper we study group-based Markov models of evolution and their mixtures. In the algebreo-geometric setting, group-based phylogenetic tree models correspond to toric varieties, while their mixtures correspond to secant and join varieties. Determining properties of these secant and join varieties can aid both in model selection and establishing parameter identifiability. Here we explore the first natural geometric property of these varieties: their dimension. The expected projective dimension of the join variety of a set of varieties is one more than the sum of their dimensions. A join variety that realizes the expected dimension is nondefective. Nondefectiveness is not only interesting from a geometric point-of-view, but has been used to establish combinatorial identifiability for several classes of phylogenetic mixture models. In this paper, we focus on group-based models where the equivalence classes of identified parameters are orbits of a subgroup of the automorphism group of the group defining the model. In particular, we show that, for these group-based models, the variety corresponding to the mixture of $r$ trees with $n$ leaves is nondefective when $n \geq 2r+5$. We also give improved bounds for claw trees and give computational evidence that 2-tree and 3-tree mixtures are nondefective for small~$n$., Comment: 24 pages, 4 figures

A double dominating set ofG is a restrained double dominating set of G if for eachx2 V (G)nS, there existsy2 V (G)nS such thatxy2 E(G). In this paper, we characterized the restrained double dominating sets in the join, and corona of two graphs. We also determine sharp bounds for the restrained double domination numbers of these graphs. In particular, we show that if G and H are any graphs without isolated vertices of orders n and m, respectively, then r 2(G H) = minfn( r(H) + 1);n 2(H)g, where r, 2, and r 2 are, respectively, the restrained domination, double domination, and restrained double domination parameters.

Given any function f : Z + &rarr, R + , let us define the f-index I f ( G ) = &sum, u &isin, V ( G ) f ( d u ) and the f-polynomial P f ( G , x ) = &sum, V ( G ) x 1 / f ( d u ) &minus, 1 , for x &gt, 0 . In addition, we define P f ( G , 0 ) = lim x &rarr, 0 + P f ( G , x ) . We use the f-polynomial of a large family of topological indices in order to study mathematical relations of the inverse degree, the generalized first Zagreb, and the sum lordeg indices, among others. In this paper, using this f-polynomial, we obtain several properties of these indices of some classical graph operations that include corona product and join, line, and Mycielskian, among others.

We introduce a join construction, as a way of completing the description of the relative conormal space of an analytic function on a complex analytic space that has a non-vanishing derivative at the origin. Then we show how to obtain a numerical criterion for Thom’s [Formula: see text] condition.

In this article we study the topology of a family of real analytic germs $$F :(\mathbb {C}^3,0) \rightarrow (\mathbb {C},0)$$ with isolated critical point at the origin, given by $$F(x,y,z)=f(x,y)\overline{g(x,y)}+z^r$$ , where $$f$$ and $$g$$ are holomorphic, $$r \in \mathbb {Z}_{}^+$$ and $$r \ge 2$$ . We describe the link $$L_F$$ as a graph manifold using its natural open book decomposition, related to the Milnor fibration of the map-germ $$f\bar{g}$$ and the description of its monodromy as a quasi-periodic diffeomorphism through its Nielsen invariants. Furthermore, such a germ $$F$$ gives rise to a Milnor fibration $$\frac{F}{|F|} :\mathbb {S}^{5} {\setminus } L_F \rightarrow \mathbb {S}^{1}$$ . We present a join theorem, which allows us to describe the homotopy type of the Milnor fibre of $$F$$ and we show some cases where the open book decomposition of $$\mathbb {S}^{5}$$ given by the Milnor fibration of $$F$$ cannot come from the Milnor fibration of a complex singularity in $$\mathbb {C}^3$$ .

In this paper, we show that the nonorientable genus of C m + C n , the join of two cycles C m and C n , is equal to $\left\lceil {\tfrac{{(m - 2)(n - 2)}} {2}} \right\rceil $ if m = 3, n ≡ 1 (mod 2), or m ≥ 4, n ≥ 4, (m, n) ≠ (4, 4). We determine that the nonorientable genus of C 4 +C 4 is 3, and that the nonorientable genus of C 3 +C n is $\tfrac{n} {2} $ if n ≡ 0 (mod 2). Our results show that a minimum nonorientable genus embedding of the complete bipartite graph K m,n cannot be extended to an embedding of the join of two cycles without increasing the genus of the surface.

The problem discussed in this paper was stated by Alexander O. Ivanov and Alexey A. Tuzhilin in 2009. It stands at the intersection of the theories of Gromov minimal fillings and Steiner minimal trees. Thus, it can be considered as one-dimensional stratified version of the Gromov minimal fillings problem. Here we state the problem; discuss various properties of one-dimensional minimal fillings, including a formula calculating their weights in terms of some special metrics characteristics of the metric spaces they join (it was obtained by A.Yu. Eremin after many fruitful discussions with participants of Ivanov-Tuzhilin seminar at Moscow State University); show various examples illustrating how one can apply the developed theory to get nontrivial results; discuss the connection with additive spaces appearing in bioinformatics and classical Steiner minimal trees having many applications, say, in transportation problem, chip design, evolution theory etc. In particular, we generalize the concept of Steiner ratio and get a few of its modifications defined by means of minimal fillings, which could give a new approach to attack the long standing Gilbert-Pollack Conjecture on the Steiner ratio of the Euclidean plane.

The problem of ring molecules and macromolecules arises in a number of contexts in physical chemistry. Perhaps the simplest example of a seven-membered loop is cycloheptane $${\rm C_{7}\rm H_{14}}$$ , which is a molecule where the carbon–carbon bonds form a regular seven-membered loop. However, it is possible to envisage much more complicated arrangements of proteins in chains comprising straight rigid sections linked in ways that enforce the same angle at all of the joins. In this paper, we present a coordinate system that reduces the problem to four free variables and three constraints. We then survey the solutions numerically and find that there are families of solutions for all join angles $${\theta}$$ between $${\pi/7}$$ and $${5\pi/7}$$ with fixed planar solutions existing for $${\theta = \pi/7}$$ , $${3\pi/7}$$ and $${5\pi/7}$$ . The available families of solutions undergo a major reorganisation at the join angle $${\theta = \pi/3}$$ , where 28 intersecting solutions form a single connected network of configurations.

Using the combinatorics of the underlying simplicial complex $K$, we give various upper and lower bounds for the Lusternik-Schnirelmann (LS) category of moment-angle complexes $\zk$. We describe families of simplicial complexes and combinatorial operations which allow for a systematic description of the LS category. In particular, we characterise the LS category of moment-angle complexes $\zk$ over triangulated $d$-manifolds $K$ for $d\leq 2$, as well as higher dimension spheres built up via connected sum, join, and vertex doubling operations. %This characterisation is given in terms of the combinatorics of $K$, the cup product length of $H^*(\zk)$, as well as a certain Massey products. We show that the LS category closely relates to vanishing of Massey products in $H^*(\zk)$ and through this connection we describe first structural properties of Massey products in moment-angel manifolds. Some of further applications include calculations of the LS category and the description of conditions for vanishing of Massey products for moment-angle manifolds over fullerenes, Pogorelov polytopes and $k$-neighbourly complexes, which double as important examples of hyperbolic manifolds., Added characterisation of LS-category of moment angle complexes over two dimensional surfaces; simplified some arguments

For the set C(X) of real-valued continuous functions on a Tychonoff space X, the compact-open topology on C(X) is a "set-open topology". This paper studies the separation and countability properties of the space C(X) having the topology given by the join of the compact-open topology and an "open-set topology" called the open-point topology, that was introduced in [6]., Comment: Some corrections are made

We consider a "twisted" noncommutative join procedure for unital $C^*$-algebras which admit actions by a compact abelian group $G$ and its discrete abelian dual $\Gamma$, so that we may investigate an analogue of Baum-Dabrowski-Hajac noncommutative Borsuk-Ulam theory in the twisted setting. Namely, under what conditions is it guaranteed that an equivariant map $\phi$ from a unital $C^*$-algebra $A$ to the twisted join of $A$ and $C^*(\Gamma)$ cannot exist? This pursuit is motivated by the twisted analogues of even spheres, which admit the same $K_0$ groups as even spheres and have an analogous Borsuk-Ulam theorem that is detected by $K_0$, despite the fact that the objects are not themselves deformations of a sphere. We find multiple sufficient conditions for twisted Borsuk-Ulam theorems to hold, one of which is the addition of another equivariance condition on $\phi$ that corresponds to the choice of twist. However, we also find multiple examples of equivariant maps $\phi$ that exist even under fairly restrictive assumptions. Finally, we consider an extension of unital contractibility (in the sense of Dabrowski-Hajac-Neshveyev) "modulo $k$.", Comment: 12 pages. Version 2 implemented the following changes: stated definitions and theorems in greater generality, cleaned up proofs, and highlighted key examples (as opposed to referring to them casually in the text)

We propose a framework to study the computational complexity of definable relations on a structure. Many of the notions we discuss are old, but the viewpoint is new. We believe that all the pieces fit together smoothly under this new point of view. We also survey related results in the area. More concretely, we study the space of sequences of relations over a given structure. On this space, we develop notions of c.e.-ness, reducibility, join and jump. These notions are equivalent to other notions studied in other settings. We explain the equivalences and differences between these notions.

The dual of the join semilattice of proper compact Scott open subsets of a domain D is called the Smyth powerdomain of D. The Smyth powerdomain is used in programming semantics as a model for demonic nondeterminism. In this paper, we introduce the concept of partial information systems; and, as an application, show that the Smyth powerdomain of any domain can be realized in terms of the sub partial information systems of the domain’s corresponding information system.