Note on unicyclic graphs with given number of pendent vertices and minimal energy

For a simple graph G, the energy E(G) is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let G(n,p) denote the set of unicyclic graphs with n vertices and p pendent vertices. In [H. Hua, M. Wang, Unicyclic graphs with given number of pendent vertices and minimal energy, Linear Algebra Appl. 426 (2007) 478-489], Hua and Wang discussed the graphs that have minimal energy in G(n,p), and determined the minimal-energy graphs among almost all different cases of n and p. In their work, certain case of the values was left as an open problem in which the minimal-energy species have to be chosen in two candidate graphs, but cannot be determined by comparing of the corresponding coefficients of their characteristic polynomials. This paper aims at solving the problem completely, by using the well-known Coulson integral formula.     

Finding all vertices of a convex polyhedron

Summary The paper describes an algorithm for finding all vertices of a convex polyhedron defined by a system of linear equations and by non-negativity conditions for variables. The algorithm is described using terminology of the theory of graphs and it seems to provide a computationally effective method. An illustrative example and some experiences with computations on a computer are given.     

A polyhedron with alls—t cuts as vertices,and adjacency of cuts

Consider the polyhedron represented by the dual of the LP formulation of the maximums–t flow problem. It is well known that the vertices of this polyhedron are integral, and can be viewed ass–t cuts in the given graph. In this paper we show that not alls–t cuts appear as vertices, and we give a characterization. We also characterize pairs of cuts that form edges of this polyhedron. Finally, we show a set of inequalities such that the corresponding polyhedron has as its vertices alls–t cuts.Work done at the Department of Computer Science and Engineering, Indian Institute of Technology, Delhi, India.     

On the maximum number of permutations with given maximal or minimal distance

Let us denote by R(k, ? λ)[R(k, ? λ)] the maximal number $\text{M}$ such that there exist $\text{M}$ different permutations of the set {1,…, k} such that any two of them have at least λ (at most λ, respectively) common positions. We prove the inequalities R(k, ? λ) ? kR(k ? 1, ? λ ? 1), R(k, ? λ) ? R(k, ? λ ? 1) ? k!, R(k, ? λ) ? kR(k ? 1, ? λ ? 1). We show: R(k, ? k ? 2) = 2, R(k, ? 1) = (k ? 1)!, R(p^m, ? 2) = (p^m ? 2)!, R(p^m + 1, ? 3) = (p^m ? 2)!, $\text{R(k, ? k ? 3) =}\text{k!}\text{2}$, R(k, ? 0) = k, R(p^m, ? 1) = p^m(p^m ? 1), R(p^m + 1, ? 2) = (p^m + 1)p^m(p^m ? 1). The exact value of R(k, ? λ) is determined whenever k ? k₀(k ? λ); we conjecture that R(k, ? λ) = (k ? λ)! for k ? k₀(λ). Bounds for the general case are given and are used to determine that the minimum of |R(k, ? λ) ? R(k, ? λ)| is attained for $\text{λ = (}\text{k}\text{2}\text{) + O(}\text{k}\text{log}\text{k}\text{)}$.     

Simple vertices of maximal minor polytopes

Themaximal minor polytope Π _{m, n} is the Newton polytope of the product of all maximal minors of anm×n matrix of indeterminates. The family of polytopes {Π _{m, n} } interpolates between the symmetric transportation polytope (form=n−1) and the permutohedron (form=2). Both transportation polytope and the permutohedron aresimple polytopes but in general Π _{m, n} is not simple. The main result of this paper is an explicit construction of a class of simple vertices of Π _{m, n} for generalm andn. We call themvertices of diagonal type. For every such vertexv we explicitly describe all the edges and facets of Π _{m, n} which containv. Simple vertices of Π _{m, n} have an interesting algebro-geometric application: they correspond tononsingular extreme toric degenerations of the determinantal variety ofm×n matrices not of full rank. Andrei Zelevinsky was partially supported by the NSF under Grant DMS-9104867.     

Algebraic surfaces with the symmetry group of the polyhedron 321

Translated from Ukrainskii Geometricheskii Sbornik, No. 31, pp. 108–112, 1988.     

On deformation with constant Milnor number and Newton polyhedron

We show that every \(\mu \)-constant family of isolated hypersurface singularities satisfying a nondegeneracy condition in the sense of Kouchnirenko, is topologically trivial, also is equimultiple.     

A characterization of maximal and minimal Fermat curves

In this article we provide a characterization of maximal and minimal Fermat curves using the classification of supersingular Fermat curves.     

Point-color-symmetric graphs with a prime number of vertices

We construct and enumerate all point-color-symmetric digraphs and graphs with a prime number of vertices. Our result unifies and generalizes the similar results for vertex transitive graphs and symmetric graphs.     

Nonflexible polyhedra with a small number of vertices

The paper shows which embedded and immersed polyhedra with no more than eight vertices are nonflexible. It turns out that all embedded polyhedra are nonflexible, except possibly for polyhedra of one of the combinatorial types, for which the problem still remains open. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 1, pp. 143–165, 2006.     

On the maximal volume of convex bodies with few vertices

The problem of determining the largest volume of a (d + 2)-point set in E^d of unit diameter is settled. The extremal polytopes are described completely.     

Circle-sum and minimal genus surfaces in ruled 4-manifolds

We describe a circle-sum construction of smoothly embedded surface in a smooth 4-manifold. We apply this construction to give a simpler solution of the minimal genus problem for nontrivial bundles over surfaces. We also treat the case of blow-ups.

     

A bound for the number of vertices of a polytope with applications

We prove that the number of vertices of a polytope of a particular kind is exponentially large in the dimension of the polytope. As a corollary, we prove that an n-dimensional centrally symmetric polytope with O(n) facets has {ie1-1} vertices and that the number of r-factors in a k-regular graph is exponentially large in the number of vertices of the graph provided k≥2r+1 and every cut in the graph with at least two vertices on each side has more than k/r edges.     

On the maximal energy tree with two maximum degree vertices

For a simple graph G, the energy E(G) is defined as the sum of the absolute values of all eigenvalues of its adjacent matrix.For Δ?3 and t?3, denote by T_a(Δ,t) (or simply T_a) the tree formed from a path P_t on t vertices by attaching Δ-1P₂’s on each end of the path P_t, and T_b(Δ,t) (or simply T_b) the tree formed from P_t+2 by attaching Δ-1P₂’s on an end of the P_t+2 and Δ-2P₂’s on the vertex next to the end.In Li et al.(2009) [16] proved that among trees of order n with two vertices of maximum degree Δ, the maximal energy tree is either the graph T_a or the graph T_b, where t=n+4-4Δ?3.However, they could not determine which one of T_a and T_b is the maximal energy tree.This is because the quasi-order method is invalid for comparing their energies.In this paper, we use a new method to determine the maximal energy tree.It turns out that things are more complicated.We prove that the maximal energy tree is T_b for Δ?7 and any t?3, while the maximal energy tree is T_a for Δ=3 and any t?3.Moreover, for Δ=4, the maximal energy tree is T_a for all t?3 but one exception that t=4, for which T_b is the maximal energy tree.For Δ=5, the maximal energy tree is T_b for all t?3 but 44 exceptions that t is both odd and 3?t?89, for which T_a is the maximal energy tree.For Δ=6, the maximal energy tree is T_b for all t?3 but three exceptions that t=3,5,7, for which T_a is the maximal energy tree.One can see that for most cases of Δ, T_b is the maximal energy tree,Δ=5 is a turning point, and Δ=3 and 4 are exceptional cases, which means that for all chemical trees (whose maximum degrees are at most 4) with two vertices of maximum degree at least 3, T_a has maximal energy, with only one exception T_a(4,4).