Bondage number of strong product of two paths

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_m?P_n) = 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).     

OD-Characterization of certain four dimensional linear groups with related results concerning degree patterns

The prime graph of a finite group G, which is denoted by GK(G), is a simple graph whose vertex set is comprised of the prime divisors of |G| and two distinct prime divisors p and q are joined by an edge if and only if there exists an element of order pq in G. Let p₁2<?<p_k be all prime divisors of |G|. Then the degree pattern of G is defined as D(G) = (degG(p₁), degG(p₂), ? , degG(p_k)), where degG(p) signifies the degree of the vertex p in GK(G). A finite group H is said to be OD-characterizable if G? H for every finite group G such that |G| = |H| and D(G) = D(H). The purpose of this article is threefold. First, it finds sharp upper and lower bounds on ?(G), the sum of degrees of all vertices in GK(G), for any finite group G (Theorem 2.1). Second, it provides the degree of vertices 2 and the characteristic p of the base field of any finite simple group of Lie type in their prime graphs (Propositions 3.1-3.7). Third, it proves the linear groups L₄(q), q = 19, 23, 27, 29, 31, 32, and 37, are OD-characterizable (Theorem 4.2).     

Largest signless Laplacian spectral radius of uniform supertrees with diameter and pendent edges (vertices)

Let \mathbb{S}(m; d; k) be the set of k-uniform supertrees with m edges and diameter d; and S₁(m; d; k) be the k-uniform supertree obtained from a loose path u₁; e₁; u₂; e₂,..., u_d; e_d; u_d+1 with length d by attaching m — d edges at vertex u_{\lfloor d/2\rfloor +1}: In this paper, we mainly determine S₁(m; d; k) with the largest signless Laplacian spectral radius in \mathbb{S}(m; d; k) for 3≤d≤m –1: We also determine the supertree with the second largest signless Laplacian spectral radius in \mathbb{S}(m; 3; k): Furthermore, we determine the unique k-uniform supertree with the largest signless Laplacian spectral radius among all k-uniform supertrees with n vertices and pendent edges (vertices).     

S-semiembedded subgroups of finite groups

A subgroup H of a finite group G is said to be s-semipermutable in G if it is permutable with every Sylow p-subgroup of G with (p, |H|) = 1. We say that a subgroup H of a finite group G is S-semiembedded in G if there exists an s-permutable subgroup T of G such that TH is s-permutable in G and T∩H≤Hs¯G, where Hs¯G is an s-semipermutable subgroup of G contained in H. In this paper, we investigate the influence of S-semiembedded subgroups on the structure of finite groups.     

Bipartite double cover and perfect 2-matching covered graph with its algorithm

Let B(G) denote the bipartite double cover of a non-bipartite graph G with v≥2 vertices and ? edges. We prove that G is a perfect 2-matching covered graph if and only if B(G) is a 1-extendable graph. Furthermore, we prove that B(G) is a minimally 1-extendable graph if and only if G is a minimally perfect 2-matching covered graph and for each e = xy ∈ E(G), there is an independent set S in G such that |Γ_G(S)| = |S| + 1, x ∈S and |Γ_G-xy(S) | = |S|. Then, we construct a digraph D from B(G) or G and show that D is a strongly connected digraph if and only if G is a perfect 2-matching covered graph. So we design an algorithm in O(v?) time that determines whether G is a perfect 2-matching covered graph or not.     

Flag-transitive 2-(v, k, λ) symmetric designs with (k, λ) = 1 and alternating socle

Consider the flag-transitive 2-(v, k, λ) symmetric designs with (k, λ) = 1. We prove that if {D} is a nontrivial 2-(v, k, λ) symmetric design with (k, λ) = 1 and G≤Aut({D}) is flag-transitive with Soc(G) = A_n for n≥5, then {D} is the projective space PG₂(3,2) and G = A₇.     

Transversality on locally pseudocompact groups

Two non-discrete Hausdorff group topologies \tau and \delta on a group G are called transversal if the least upper bound \tau \vee \delta of \tau and \delta is the discrete topology. In this paper, we discuss the existence of transversal group topologies on locally pseudocompact, locally precompact, or locally compact groups. We prove that each locally pseudocompact, connected topological group satisfies central subgroup paradigm, which gives an affrmative answer to a problem posed by Dikranjan, Tkachenko, and Yaschenko [Topology Appl., 2006, 153:3338-3354]. For a compact normal subgroup K of a locally compact totally disconnected group G, if G admits a transversal group topology, then G/K admits a transversal group topology, which gives a partial answer again to a problem posed by Dikranjan, Tkachenko, and Yaschenko in 2006. Moreover, we characterize some classes of locally compact groups that admit transversal group topologies.     

Group inverses for some 2 × 2 block matrices over rings

We first consider the group inverses of the block matrices (A0BC) over a weakly finite ring. Then we study the sufficient and necessary conditions for the existence and the representations of the group inverses of the block matrices (ACBD) over a ring with unity 1 under the following conditions respectively: (i) B = C, D = 0, B^# and (B^πA) ^# both exist; (ii) B is invertible and m = n; (iii) A^# and (D - CA^#B)^# both exist, C = CAA^# , where A and D are m × m and n × n matrices, respectively.     

Property T and strong property T for unital*-homomorphisms

We introduce and study property T and strong property T for unital*-homomorphisms between two unital C^*-algebras.We also consider the relations between property T and invariant subspaces for some canonical unital^-representations.As a corollary,we show that when G is a discrete group,G is finite if and only if G is amenable and the inclusion map i:Cr^*(G)→B(l^2(G))has property T.We also give some new equivalent forms of property T for countable discrete groups and strong property T for unital C^*-algebras.     

G-stable support τ-tilting modules

Motivated by τ-tilting theory developed by T. Adachi, O. Iyama, I. Reiten, for a nite-dimensional algebra Λwith action by a nite group G; we introduce the notion of G-stable support τ-tilting modules. Then we establish bijections among G-stable support τ-tilting modules over Λ; G-stable two-term silting complexes in the homotopy category of bounded complexes of nitely generated projective Λ-modules, and G-stable functorially nite torsion classes in the category of nitely generated left Λ-modules. In the case when Λ is the endomorphism of a G-stable cluster-tilting object T over a Hom-nite 2-Calabi-Yau triangulated category ℓ with a G-action, these are also in bijection with G-stable cluster-tilting objects in ℓ : Moreover, we investigate the relationship between stable support τ-tilitng modules over Λ and the skew group algebra ΛG:     

A combinatorial approach to doubly transitive permutation groups

Let G be a doubly but not triply transitive group on a set X. We give an algorithm to construct the orbits of G acting on X×X×X by combining those of its stabilizer H of a point of X If the group H is given first, we compute the orbits of its transitive extension G, if it exists. We apply our algorithm to G=PSL(m,q) and Sp(2m,2), m3, successfully. We go forward to compute the transitive extension of G itself. In our construction we use a superscheme defined by the orbits of H on X×X×X and do not use group elements.     

Sum-connectivity index of a graph

Let G be a simple connected graph, and let di be the degree of its i-th vertex. The sum-connectivity index of the graph G is defined as χ(G)=Σvivj∈E(G)? (di+dj)−1/2. We discuss the effect on χ(G) of inserting an edge into a graph. Moreover, we obtain the relations between sum-connectivity index and Randić index.     

Distance signless Laplacian eigenvalues of graphs

Suppose that the vertex set of a graph G is V(G)=\{v\mathrm{1},v\mathrm{2},\mathrm{...},vn\}. The transmission Tr(vi) (or D_i) of vertex vi is defined to be the sum of distances from vi to all other vertices. Let Tr(G) be the n\times n diagonal matrix with its (i, i)-entry equal to {{\displaystyle Tr}}_G(vi). The distance signless Laplacian spectral radius of a connected graph G is the spectral radius of the distance signless Laplacian matrix of G, defined as L(G)=Tr(G)+D(G), where D(G) is the distance matrix of G. In this paper, we give a lower bound on the distance signless Laplacian spectral radius of graphs and characterize graphs for which these bounds are best possible. We obtain a lower bound on the second largest distance signless Laplacian eigenvalue of graphs. Moreover, we present lower bounds on the spread of distance signless Laplacian matrix of graphs and trees, and characterize extremal graphs.     

On ramsey numbers for large disjoint unions of graphs

Let be a fixed finite set of connected graphs. Results are given which, in principle, permit the Ramsey number r(G, H) to be evaluated exactly when G and H are sufficiently large disjoint unions of graphs taken from . Such evaluations are often possible in practice, as shown by several examples. For instance, when m and n are large, and mn,

r(mK_k, nK_l)=(k − 1)m+ln+r(K_k−1, K_l−1)−2.

     

Maximum genus, connectivity and minimal degree of graphs

This paper is devoted to the lower bounds on the maximum genus of graphs. A simple statement of our results in this paper can be expressed in the following form:

Let G be a k-edge-connected graph with minimum degree δ, for each positive integer k(3), there exists a non-decreasing function f(δ) such that the maximum genus γ_M(G) of G satisfies the relation γ_M(G)f(δ)β(G), and furthermore that lim_δ→∞f(δ)=1/2, where β(G)=|E(G)|-|V(G)|+1 is the cycle rank of G.

The result shows that lower bounds of the maximum genus of graphs with any given connectivity become larger and larger as their minimum degree increases, and complements recent results of several authors.     

Quasi-periodic solutions for class of Hamiltonian partial differential equations with fixed constant potential

We consider Hamiltonian partial differential equations u_tt +|∂_x|u+ σu = f(u), x ∈ T, t ∈?, with periodic boundary conditions, where f(u) is a real-analytic function of the form f(u) = u⁵ + o(u⁵) near u = 0, σ ∈ (0, 1) is a fixed constant, and \mathbb{T}=?/\mathrm{2}\pi \mathbb{Z}T= R/2πZ. A family of quasi-periodic solutions with 2-dimensional are constructed for the equation above with σ ∈ (0, 1)\ ?. The proof is based on infinite-dimensional KAM theory and partial Birkhoff normal form.     

Edge-coloring of multigraphs

We introduce a monotone invariant π(G) on graphs and show that it is an upper bound of the chromatic index of graphs. Moreover, there exist polynomial time algorithms for computing π(G) and for coloring edges of a multigraph G by π(G) colors. This generalizes the classical edge-coloring theorems of Shannon and Vizing.     

Wintgen ideal submanifolds with a low-dimensional integrable distribution

Submanifolds in space forms satisfy the well-known DDVV inequality. A submanifold attaining equality in this inequality pointwise is called a Wintgen ideal submanifold. As conformal invariant objects, Wintgen ideal submanifolds are investigated in this paper using the framework of M?bius geometry. We classify Wintgen ideal submanfiolds of dimension m≥3 and arbitrary codimension when a canonically defined 2-dimensional distribution ?2 is integrable. Such examples come from cones, cylinders, or rotational submanifolds over super-minimal surfaces in spheres, Euclidean spaces, or hyperbolic spaces, respectively. We conjecture that if ?2 generates a k-dimensional integrable distribution ?kand k<m, then similar reduction theorem holds true. This generalization when k = 3 has been proved in this paper.