Upper Bounds for the Paired-Domination Numbers of Graphs

A set \(S\subseteq V\) is a paired-dominating set if every vertex in \(V{\setminus } S\) has at least one neighbor in S and the subgraph induced by S contains a perfect matching. The paired-domination number of a graph G, denoted by \(\gamma _{pr}(G)\), is the minimum cardinality of a paired-dominating set of G. A conjecture of Goddard and Henning says that if G is not the Petersen graph and is a connected graph of order n with minimum degree \(\delta (G)\ge 3\), then \(\gamma _{pr}(G)\le 4n/7\). In this paper, we confirm this conjecture for k-regular graphs with \(k\ge 4\).     

On eigenvalues of Seidel matrices and Haemers’ conjecture

For a graph G, let S(G) be the Seidel matrix of G and \({\theta }_1(G),\ldots ,{\theta }_n(G)\) be the eigenvalues of S(G). The Seidel energy of G is defined as \(|{\theta }_1(G)|+\cdots +|{\theta }_n(G)|\). Willem Haemers conjectured that the Seidel energy of any graph with n vertices is at least \(2n-2\), the Seidel energy of the complete graph with n vertices. Motivated by this conjecture, we prove that for any \(\alpha \) with \(0<\alpha <2,|{\theta }_1(G)|^\alpha +\cdots +|{\theta }_n(G)|^\alpha \geqslant (n-1)^\alpha +n-1\) if and only if \(|\hbox {det}\,S(G)|\geqslant n-1\). This, in particular, implies the Haemers’ conjecture for all graphs G with \(|\hbox {det}\,S(G)|\geqslant n-1\). A computation on the fraction of graphs with \(|\hbox {det}\,S(G)|<n-1\) is reported. Motivated by that, we conjecture that almost all graphs G of order n satisfy \(|\hbox {det}\,S(G)|\geqslant n-1\). In connection with this conjecture, we note that almost all graphs of order n have a Seidel energy of order \(\Theta (n^{3/2})\). Finally, we prove that self-complementary graphs G of order \(n\equiv 1\pmod 4\) have \(\det S(G)=0\).     

A method to find generators of a semi-simple Lie group via the topology of its flag manifolds

In this paper we continue to develop the topological method to get semigroup generators of semi-simple Lie groups. Consider a subset \(\Gamma \subset G\) that contains a semi-simple subgroup \(G_{1}\) of G. If one can show that \( \Gamma \) does not leave invariant a contractible subset on any flag manifold of G, then \(\Gamma \) generates G if \(\mathrm {Ad}\left( \Gamma \right) \) generates a Zariski dense subgroup of the algebraic group \(\mathrm {Ad}\left( G\right) \). The proof is reduced to check that some specific closed orbits of \(G_{1}\) in the flag manifolds of G are not trivial in the sense of algebraic topology. Here, we consider three different cases of semi-simple Lie groups G and subgroups \(G_{1}\subset G\).     

The Dirichlet problem for Einstein metrics on cohomogeneity one manifolds

Let \(G{/}H\) be a compact homogeneous space, and let \(\hat{g}_0\) and \(\hat{g}_1\) be G-invariant Riemannian metrics on \(G/H\). We consider the problem of finding a G-invariant Einstein metric g on the manifold \(G/H\times [0,1]\) subject to the constraint that g restricted to \(G{/}H\times \{0\}\) and \(G/H\times \{1\}\) coincides with \(\hat{g}_0\) and \(\hat{g}_1\), respectively. By assuming that the isotropy representation of \(G/H\) consists of pairwise inequivalent irreducible summands, we show that we can always find such an Einstein metric.     

The non-abelian tensor square of residually finite groups

Let m, n be positive integers and p a prime. We denote by \(\nu (G)\) an extension of the non-abelian tensor square \(G \otimes G\) by \(G \times G\). We prove that if G is a residually finite group satisfying some non-trivial identity \(f \equiv ~1\) and for every \(x,y \in G\) there exists a p-power \(q=q(x,y)\) such that \([x,y^{\varphi }]^q = 1\), then the derived subgroup \(\nu (G)'\) is locally finite (Theorem A). Moreover, we show that if G is a residually finite group in which for every \(x,y \in G\) there exists a p-power \(q=q(x,y)\) dividing \(p^m\) such that \([x,y^{\varphi }]^q\) is left n-Engel, then the non-abelian tensor square \(G \otimes G\) is locally virtually nilpotent (Theorem B).     

Packing chromatic number versus chromatic and clique number

The packing chromatic number \(\chi _{\rho }(G)\) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets \(V_i\), \(i\in [k]\), where each \(V_i\) is an i-packing. In this paper, we investigate for a given triple (a, b, c) of positive integers whether there exists a graph G such that \(\omega (G) = a\), \(\chi (G) = b\), and \(\chi _{\rho }(G) = c\). If so, we say that (a, b, c) is realizable. It is proved that \(b=c\ge 3\) implies \(a=b\), and that triples \((2,k,k+1)\) and \((2,k,k+2)\) are not realizable as soon as \(k\ge 4\). Some of the obtained results are deduced from the bounds proved on the packing chromatic number of the Mycielskian. Moreover, a formula for the independence number of the Mycielskian is given. A lower bound on \(\chi _{\rho }(G)\) in terms of \(\Delta (G)\) and \(\alpha (G)\) is also proved.     

Packing Chromatic Number of Base-3 Sierpiński Graphs

The packing chromatic number \(\chi _{\rho }(G)\) of a graph G is the smallest integer k such that there exists a k-vertex coloring of G in which any two vertices receiving color i are at distance at least \(i+1\). Let \(S^n\) be the base-3 Sierpiński graph of dimension n. It is proved that \(\chi _{\rho }(S^1) = 3\), \(\chi _{\rho }(S^2) = 5\), \(\chi _{\rho }(S^3) = \chi _{\rho }(S^4) = 7\), and that \(8\le \chi _\rho (S^n) \le 9\) holds for any \(n\ge 5\).     

A characterization of the desarguesian and the Figueroa planes of order $$q^{3}$$

Let \(\Pi \) be a plane of order \(q^{3}\), \(q>2\), admitting \(G\cong PGL(3,q)\) as a collineation group. By Dempwolff (Geometriae Dedicata 18:101–112, 1985) the plane \(\Pi \) contains a G-invariant subplane \(\pi _{0}\) isomorphic to PG(2, q) on which G acts 2-transitively. In this paper it is shown that, if the homologies of \(\pi _{0}\) contained in G extend to \(\Pi \) then \(\Pi \) is either the desarguesian or the Figueroa plane.     

On the density of images of the power maps in Lie groups

Let G be a connected Lie group. In this paper, we study the density of the images of individual power maps \(P_k:G\rightarrow G:g\mapsto g^k\). We give criteria for the density of \(P_k(G)\) in terms of regular elements, as well as Cartan subgroups. In fact, we prove that if \(\mathrm{Reg}(G)\) is the set of regular elements of G, then \(P_k(G)\cap \mathrm{Reg}(G)\) is closed in \(\mathrm{Reg}(G)\). On the other hand, the weak exponentiality of G turns out to be equivalent to the density of all the power maps \(P_k\). In linear Lie groups, weak exponentiality reduces to the density of \(P_2(G)\). We also prove that the density of the image of \(P_k\) for G implies the same for any connected full rank subgroup.     

Bochner representable operators on Banach function spaces

Let \((E,\Vert \cdot \Vert _E)\) be a Banach function space, \(E'\) the Köthe dual of E and \((X,\Vert \cdot \Vert _X)\) be a Banach space. It is shown that every Bochner representable operator \(T:E\rightarrow X\) maps relatively \(\sigma (E,E')\)-compact sets in E onto relatively norm compact sets in X. If, in particular, the associated norm \(\Vert \cdot \Vert _{E'}\) on \(E'\) is order continuous, then every Bochner representable operator \(T:E\rightarrow X\) is \((\gamma _E,\Vert \cdot \Vert _X)\)-compact, where \(\gamma _E\) stands for the natural mixed topology on E. Applications to Bochner representable operators on Orlicz spaces are given.     

Homomorphism reductions on Polish groups

In an earlier paper, we introduced the following pre-order on the subgroups of a given Polish group: if G is a Polish group and \(H,L \subseteq G\) are subgroups, we say H is homomorphism reducible to L iff there is a continuous group homomorphism \(\varphi : G \rightarrow G\) such that \(H = \varphi ^{-1} (L)\). We previously showed that there is a \(K_\sigma \) subgroup L of the countable power of any locally compact Polish group G such that every \(K_\sigma \) subgroup of \(G^\omega \) is homomorphism reducible to L. In the present work, we show that this fails in the countable power of the group of increasing homeomorphisms of the unit interval.     

Relationships Between the 2-Metric Dimension and the 2-Adjacency Dimension in the Lexicographic Product of Graphs

Given a connected simple graph \(G=(V(G),E(G))\), a set \(S\subseteq V(G)\) is said to be a 2-metric generator for G if and only if for any pair of different vertices \(u,v\in V(G)\), there exist at least two vertices \(w_1,w_2\in S\) such that \(d_G(u,w_i)\ne d_G(v,w_i)\), for every \(i\in \{1,2\}\), where \(d_G(x,y)\) is the length of a shortest path between x and y. The minimum cardinality of a 2-metric generator is the 2-metric dimension of G, denoted by \(\dim _2(G)\). The metric \(d_{G,2}: V(G)\times V(G)\longmapsto {\mathbb {N}}\cup \{0\}\) is defined as \(d_{G,2}(x,y)=\min \{d_G(x,y),2\}\). Now, a set \(S\subseteq V(G)\) is a 2-adjacency generator for G, if for every two vertices \(x,y\in V(G)\) there exist at least two vertices \(w_1,w_2\in S\), such that \(d_{G,2}(x,w_i)\ne d_{G,2}(y,w_i)\) for every \(i\in \{1,2\}\). The minimum cardinality of a 2-adjacency generator is the 2-adjacency dimension of G, denoted by \({\mathrm {adim}}_2(G)\). In this article, we obtain closed formulae for the 2-metric dimension of the lexicographic product \(G\circ H\) of two graphs G and H. Specifically, we show that \(\dim _2(G\circ H)=n\cdot {\mathrm {adim}}_2(H)+f(G,H),\) where \(f(G,H)\ge 0\), and determine all the possible values of f(G, H).     

Vizing Bound for the Chromatic Number on Some Graph Classes

We are interested in hereditary classes of graphs \({\mathcal {G}}\) such that every graph \(G \in {\mathcal {G}}\) satisfies \(\varvec{\chi }(G) \le \omega (G) + 1\), where \(\chi (G)\) (\(\omega (G)\)) denote the chromatic (clique) number of G. This upper bound is called the Vizing bound for the chromatic number. Apart from perfect graphs few classes are known to satisfy the Vizing bound in the literature. We show that if G is (\(P_6, S_{1, 2, 2}\), diamond)-free, then \(\chi (G) \le \omega (G)+1\), and we give examples to show that the bound is sharp.     

Some infinite families of Ramsey ($${\varvec{P}}_\mathbf{3},{\varvec{P}}_{{\varvec{n}}}$$)-minimal trees

For any given two graphs G and H, the notation \(F\rightarrow \) (G, H) means that for any red–blue coloring of all the edges of F will create either a red subgraph isomorphic to G or a blue subgraph isomorphic to H. A graph F is a Ramsey (G, H)-minimal graph if \(F\rightarrow \) (G, H) but \(F-e\nrightarrow (G,H)\), for every \(e \in E(F)\). The class of all Ramsey (G, H)-minimal graphs is denoted by \(\mathcal {R}(G,H)\). In this paper, we construct some infinite families of trees belonging to \(\mathcal {R}(P_3,P_n)\), for \(n=8\) and 9. In particular, we give an algorithm to obtain an infinite family of trees belonging to \(\mathcal {R}(P_3,P_n)\), for \(n\ge 10\).     

On the eigenvalues and spectral radius of starlike trees

Let \(k\ge 1\) and \(n_1,\ldots ,n_k\ge 1\) be some integers. Let \(S(n_1,\ldots ,n_k)\) be a tree T such that T has a vertex v of degree k and \(T{\setminus } v\) is the disjoint union of the paths \(P_{n_1},\ldots ,P_{n_k}\), that is \(T{\setminus } v\cong P_{n_1}\cup \cdots \cup P_{n_k}\) so that every neighbor of v in T has degree one or two. The tree \(S(n_1,\ldots ,n_k)\) is called starlike tree, a tree with exactly one vertex of degree greater than two, if \(k\ge 3\). In this paper we obtain the eigenvalues of starlike trees. We find some bounds for the largest eigenvalue (for the spectral radius) of starlike trees. In particular we prove that if \(k\ge 4\) and \(n_1,\ldots ,n_k\ge 2\), then \(\frac{k-1}{\sqrt{k-2}}<\lambda _1(S(n_1,\ldots ,n_k))<\frac{k}{\sqrt{k-1}}\), where \(\lambda _1(T)\) is the largest eigenvalue of T. Finally we characterize all starlike trees that all of whose eigenvalues are in the interval \((-2,2)\).     

Connected size Ramsey number for matchings vs. small stars or cycles

The notation \(F\rightarrow (G,H)\) means that if the edges of F are colored red and blue, then the red subgraph contains a copy of G or the blue subgraph contains a copy of H. The connected size Ramsey number \(\hat{r}_c(G,H)\) of graphs G and H is the minimum size of a connected graph F satisfying \(F\rightarrow (G,H)\). For \(m \ge 2,\) the graph consisting of m independent edges is called a matching and is denoted by \(mK_2\). In 1981, Erdös and Faudree determined the size Ramsey numbers for the pair \((mK_2, K_{1,t})\). They showed that the disconnected graph \(mK_{1,t} \rightarrow (mK_2,K_{1,t})\) for \( t,m \ge 1\). In this paper, we will determine the connected size Ramsey number \(\hat{r}_c(nK_2, K_{1,3})\) for \(n\ge 2\) and \(\hat{r}_c(3K_2, C_4)\). We also derive an upper bound of the connected size Ramsey number \(\hat{r}_c(nK_2, C_4),\) for \(n\ge 4\).     

Generating Hyperbolic Singularities in Semitoric Systems Via Hopf Bifurcations

Let \((M,\Omega )\) be a connected symplectic 4-manifold and let \(F=(J,H) :M\rightarrow \mathbb {R}^2\) be a completely integrable system on M with only non-degenerate singularities. Assume that F does not have singularities with hyperbolic blocks and that \(p_1,\ldots ,p_n\) are the focus–focus singularities of F. For each subset \(S=\{i_1,\ldots ,i_j\}\), we will show how to modify F locally around any \(p_i, i \in S\), in order to create a new integrable system \(\widetilde{F}=(J, \widetilde{H}) :M \rightarrow \mathbb {R}^2\) such that its classical spectrum \(\widetilde{F}(M)\) contains j smooth curves of singular values corresponding to non-degenerate transversally hyperbolic singularities of \(\widetilde{F}\). Moreover the focus–focus singularities of \(\widetilde{F}\) are precisely \(p_i\), \(i \in \{1,\ldots ,n\} \setminus S\). The proof is based on Eliasson’s linearization theorem for non-degenerate singularities, and properties of the Hamiltonian Hopf bifurcation.     

Green’s function for elliptic systems: existence and Delmotte–Deuschel bounds

This paper is divided into two parts: In the main deterministic part, we prove that for an open domain \(D \subset \mathbb {R}^d\) with \(d \ge 2\), for every (measurable) uniformly elliptic tensor field a and for almost every point \(y \in D\), there exists a unique Green’s function centred in y associated to the vectorial operator \(-\nabla \cdot a\nabla \) in D. This result implies the existence of the fundamental solution for elliptic systems when \(d>2\), i.e. the Green function for \(-\nabla \cdot a\nabla \) in \(\mathbb {R}^d\). In the second part, we introduce a shift-invariant ensemble \(\langle \cdot \rangle \) over the set of uniformly elliptic tensor fields, and infer for the fundamental solution G some pointwise bounds for \(\langle |G(\cdot ; x,y)|\rangle \), \(\langle |\nabla _x G(\cdot ; x,y)|\rangle \) and \(\langle |\nabla _x\nabla _y G(\cdot ; x,y)|\rangle \). These estimates scale optimally in space and provide a generalisation to systems of the bounds obtained by Delmotte and Deuschel for the scalar case.     

The distortion dimension of $$\mathbb $$-rank 1 lattices

Let \(X=G/K\) be a symmetric space of noncompact type and rank \(k\ge 2\). We prove that horospheres in X are Lipschitz \((k-2)\)-connected if their centers are not contained in a proper join factor of the spherical building of X at infinity. As a consequence, the distortion dimension of an irreducible \(\mathbb {Q}\)-rank-1 lattice \(\Gamma \) in a linear, semisimple Lie group G of \(\mathbb R\)-rank k is \(k-1\). That is, given \(m< k-1\), a Lipschitz m-sphere S in (a polyhedral complex quasi-isometric to) \(\Gamma \), and a \((m+1)\)-ball B in X (or G) filling S, there is a \((m+1)\)-ball \(B'\) in \(\Gamma \) filling S such that \({{\mathrm{vol}}}B'\sim {{\mathrm{vol}}}B\). In particular, such arithmetic lattices satisfy Euclidean isoperimetric inequalities up to dimension \(k-1\).     

Edge Roman Domination on Graphs

An edge Roman dominating function of a graph G is a function \(f:E(G) \rightarrow \{0,1,2\}\) satisfying the condition that every edge e with \(f(e)=0\) is adjacent to some edge \(e'\) with \(f(e')=2\). The edge Roman domination number of G, denoted by \(\gamma '_R(G)\), is the minimum weight \(w(f) = \sum _{e\in E(G)} f(e)\) of an edge Roman dominating function f of G. This paper disproves a conjecture of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi and Sadeghian Sadeghabad stating that if G is a graph of maximum degree \(\Delta \) on n vertices, then \(\gamma _R'(G) \le \lceil \frac{\Delta }{\Delta +1} n \rceil \). While the counterexamples having the edge Roman domination numbers \(\frac{2\Delta -2}{2\Delta -1} n\), we prove that \(\frac{2\Delta -2}{2\Delta -1} n + \frac{2}{2\Delta -1}\) is an upper bound for connected graphs. Furthermore, we provide an upper bound for the edge Roman domination number of k-degenerate graphs, which generalizes results of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi and Sadeghian Sadeghabad. We also prove a sharp upper bound for subcubic graphs. In addition, we prove that the edge Roman domination numbers of planar graphs on n vertices is at most \(\frac{6}{7}n\), which confirms a conjecture of Akbari and Qajar. We also show an upper bound for graphs of girth at least five that is 2-cell embeddable in surfaces of small genus. Finally, we prove an upper bound for graphs that do not contain \(K_{2,3}\) as a subdivision, which generalizes a result of Akbari and Qajar on outerplanar graphs.