Long paths and cycles in subgraphs of the cube

Let Q _n denote the graph of the n-dimensional cube with vertex set {0, 1} ⁿ in which two vertices are adjacent if they differ in exactly one coordinate. Suppose G is a subgraph of Q _n with average degree at least d. How long a path can we guarantee to find in G? Our aim in this paper is to show that G must contain an exponentially long path. In fact, we show that if G has minimum degree at least d then G must contain a path of length 2 ^d ? 1. Note that this bound is tight, as shown by a d-dimensional subcube of Q _n . We also obtain the slightly stronger result that G must contain a cycle of length at least 2 ^d .     

The Domination Polynomial of a Graph at −1

Let G be a simple graph. The domination polynomial of a graph G of order n is the polynomial ${D(G,x)=\sum_{i=0}^{n} d(G,i) x^{i}}$ , where d(G, i) is the number of dominating sets of G of size i. In this article we investigate the domination polynomial at ?1. We give a construction showing that for each odd number n there is a connected graph G with D(G, ?1) = n.     

On quasigroups rich in associative triples

Let G be a group and $\text{G(1)}$ a quasigroup on the same underlying set. Let dist($\text{G, G(1)}$) denote the number of pairs (x, y) ?G² such that $\text{xy ≠ x 1 y}$. For a finite quasigroup Q, n = card(Q), let t = dist(Q) = min dist(G, Q), where G runs through all groups with the same underlying set, and s = s(Q) the number of non-associative triples. Then 4tn?2t²?24t?s?4tn. If 1 ? s < 3n²/32, then 3tn < s holds as well. Let n ? 168 be an even integer and let σ = min s(Q), where Q runs through all non-associative quasigroups of order n. Then σ = 16n?64.     

A maximum degree theorem for diameter-2-critical graphs

A graph is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. Let G be a diameter-2-critical graph of order n. Murty and Simon conjectured that the number of edges in G is at most ?n ²/4? and that the extremal graphs are the complete bipartite graphs K _?n/2?,?n/2?. Fan [Discrete Math. 67 (1987), 235–240] proved the conjecture for n ≤ 24 and for n = 26, while Füredi [J. Graph Theory 16 (1992), 81–98] proved the conjecture for n > n ₀ where n ₀ is a tower of 2’s of height about 10¹⁴. The conjecture has yet to be proven for other values of n. Let Δ denote the maximum degree of G. We prove the following maximum degree theorems for diameter-2-critical graphs. If Δ ≥ 0.7 n, then the Murty-Simon Conjecture is true. If n ≥ 2000 and Δ ≥ 0.6789 n, then the Murty-Simon Conjecture is true.     

Cogrowth and amenability of discrete groups

Let G be a group and g₁,…, g_t a set of generators. There are approximately (2t ? 1)ⁿ reduced words in g₁,…, g_t, of length ?n. Let $\text{}\text{?}\text{gg}{}_{\mathrm{n}}$ be the number of those which represent 1_G. We show that $\text{γ =}\text{lim}{}_{\mathrm{n\; \to \; \infty }}\text{(}\text{}\text{?}\text{gg}{}_{\mathrm{n}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}$ exists. Clearly 1 ? γ ? 2t ? 1. $\text{η =}\text{(}\text{log}\text{γ)}\text{(}\text{log}\text{(2t ? 1))}$ is the cogrowth. 0 ? η ? 1. In fact $\text{η ∈ {0} ∪ (}\text{1}\text{2}\text{, 1¦}$. The entropic dimension of G is shown to be 1 ? η. It is then proved that d(G) = 1 if and only if G is free on g₁,…, g_t and d(G) = 0 if and only if G is amenable.     

Sharp bounds of the zeroth-order general Randi? index of bicyclic graphs with given pendent vertices

Let G be a simple connected graph and α be a given real number. The zeroth-order general Randi? index of ⁰R_α(G) is defined as ∑_v∈V(G)[d_G(v)]^α, where d_G(v) denotes the degree of the vertex v of G. In this paper, for any α(≠0,1), we give sharp bounds of the zeroth-order general Randi? index ⁰R_α of all bicyclic graphs with n vertices and k pendent vertices.     

Similarity classes of3 × 3 matrices over a discrete valuation ring

Let Q be a complete discrete valuation ring. Let Π be a prime element in Q. Write P = ΠQ. For n = 1,2,…, letQ_n be the factor ring Q | Pⁿ. Let G = G1₃(Q_n. Denote by M?_n the G-module of 3 × 3 matrices over Q_n modulo scalar matrices. Canonical forms are found for every element in M?_n, and it is shown that there exist five sets of similarity classes. Some results about the general case of NxN matrices over Q also are proved.     

The extremal graph problem of the icosahedron

P. Turán has asked the following question:Let I¹² be the graph determined by the vertices and edges of an icosahedron. What is the maximum number of edges of a graph Gⁿ of n vertices if Gⁿ does not contain I¹² as a subgraph?We shall answer this question by proving that if n is sufficiently large, then there exists only one graph having maximum number of edges among the graphs of n vertices and not containing I¹². This graph Hⁿ can be defined in the following way:Let us divide n ? 2 vertices into 3 classes each of which contains [$\text{(n?2)}\text{3}$] or $\text{[}\text{(n?2)}\text{3}\text{] + 1}$ vertices. Join two vertices iff they are in different classes. Join two vertices outside of these classes to each other and to every vertex of these three classes.     

On stress matrices of (d + 1)-lateration frameworks in general position

Let (G, P) be a bar framework of n vertices in general position in ${\mathbb{R}^d}$ , for d ≤ n ? 1, where G is a (d + 1)-lateration graph. In this paper, we present a constructive proof that (G, P) admits a positive semidefinite stress matrix with rank (n ? d ? 1). We also prove a similar result for a sensor network, where the graph consists of m(≥ d + 1) anchors.     

More on “Connected (n, m)-graphs with minimum and maximum zeroth-order general Randi? index”

Let G be a graph and d(u) denote the degree of a vertex u in G. The zeroth-order general Randi? index ⁰R_α(G) of the graph G is defined as ∑_u∈V(G)d(u)^α, where the summation goes over all vertices of G and α is an arbitrary real number. In this paper we correct the proof of the main Theorem 3.5 of the paper by Hu et al. [Y. Hu, X. Li, Y. Shi, T. Xu, Connected (n,m)-graphs with minimum and maximum zeroth-order general Randi? index, Discrete Appl. Math. 155 (8) (2007) 1044-1054] and give a more general Theorem. We finally characterize ¹ for α<0 the connected G(n,m)-graphs with maximum value ⁰R_α(G(n,m)), where G(n,m) is a simple connected graph with n vertices and m edges.     

Chromatic number, clique subdivisions, and the conjectures of Hajós and Erdős-Fajtlowicz

For a graph G, let χ(G) denote its chromatic number and σ(G) denote the order of the largest clique subdivision in G. Let H(n) be the maximum of χ(G)=σ(G) over all n-vertex graphs G. A famous conjecture of Hajós from 1961 states that σ(G) ≥ χ(G) for every graph G. That is, H(n)≤1 for all positive integers n. This conjecture was disproved by Catlin in 1979. Erd?s and Fajtlowicz further showed by considering a random graph that H(n)≥cn ^1/2/logn for some absolute constant c>0. In 1981 they conjectured that this bound is tight up to a constant factor in that there is some absolute constant C such that χ(G)=σ(G) ≤ Cn ^1/2/logn for all n-vertex graphs G. In this paper we prove the Erd?s-Fajtlowicz conjecture. The main ingredient in our proof, which might be of independent interest, is an estimate on the order of the largest clique subdivision which one can find in every graph on n vertices with independence number α.     

Finding Well-Conditioned Similarities to Block-Diagonalize Nonsymmetric Matrices Is NP-Hard

Given an upper triangular matrix A ∈ R^n×n and a tolerance τ, we show that the problem of finding a similarity transformation G such that G⁻¹AG is block diagonal with the condition number of G being at most τ is NP-hard. Let ƒ(n) be a polynomial in n. We also show that the problem of finding a similarity transformation G such that G⁻¹AG is block-diagonal with the condition number of G being at most ƒ(n) times larger than the smallest possible is NP-hard.     

Inversion formulas for complex radon transform on projective varieties and boundary value problems for systems of linear PDEs

Let G ? ?P ⁿ be a linearly convex compact set with smooth boundary, D = ?P ⁿ \ G, and let D* ? (?P ⁿ )* be the dual domain. Then for an algebraic, not necessarily reduced, complete intersection subvariety V of dimension d we construct an explicit inversion formula for the complex Radon transform R _V : H ^d,d?1(V ∩ D) → H ^1,0(D*) and explicit formulas for solutions of an appropriate boundary value problem for the corresponding system of differential equations with constant coefficients on D*.     

A note on bipartite subgraphs of triangle-free graphs

Let G be a triangle-free graph on n points with m edges and vertex degrees d₁, d₂,…, d_n. Let k be the maximum number of edges in a bipartite subgraph of G. In this note we show that k ? m/2 + Σ √d_i. It follows as a corollary that k ? m/2 + cm^3/4.     

An algorithmic proof of Tutte's f-factor theorem

Let G be a multigraph on n vertices, possibly with loops. An f-factor is a subgraph of G with degree f_i at the ith vertex for i = 1, 2,…, n. Tutte's f-factor theorem is proved by providing an algorithm that either finds an f-factor or shows that it does not exist and does this in O(n³) operations. Note that the complexity bound is independent of the number of edges of G and the degrees f_i. The algorithm is easily altered to handle the problem of looking for a symmetric integral matrix with given row and column sums by assigning loops degree one. A (g,f)-factor is a subgraph of G with degree d_i at the ith vertex, where g_i ? d_i ? f_i, for i = 1,2,…, n. Lovasz's (g,f)-factor theorem is proved by providing an O(n³) algorithm to either find a (g,f)-factor or show one does not exist.     

Nilpotency indices,degrees of iterations of affine triangular automorphisms,and Schubert calculus

Let f be a triangular automorphism of the affine N-space of degree d with Jacobian 1 over a \({\mathbb{Q}}\) -algebra R. We introduce a weighted nilpotency index ν(f) for f, and give a bound of deg(f ⁿ ) in terms of N, d and ν(f) for all \({n \in \mathbb{Z}}\) . When N = 2, our formula, combined with computation of the Hilbert series of certain graded algebras, yields the estimate deg(f ⁿ ) ≤ d ² ? d + 1 for all \({n \in \mathbb{Z}}\) . If n varies through all integers, this estimate turns out to be sharp and is related, somewhat unexpectedly, to the Schubert calculus on the Grassmannian G(d ? 1, 2d ? 2). Numerical computation for small degrees suggests that this estimate restricted to the inverse degree (i.e. n = ?1) is also sharp if d ≥ 3.     

2-Cell Embeddings with Prescribed Face Lengths and Genus

Let n be a positive integer, let d ₁, . . . , d _n be a sequence of positive integers, and let ${{q = \frac{1}{2}\sum^{n}_{i=1} d_{i}\cdot}}$ . It is shown that there exists a connected graph G on n vertices, whose degree sequence is d ₁, . . . , d _n and such that G admits a 2-cell embedding in every closed surface whose Euler characteristic is at least n ? q?+?1, if and only if q is an integer and q ?? n ? 1. Moreover, the graph G can be required to be loopless if and only if d _i ?? q for i = 1, . . . , n. This, in particular, answers a question of Skopenkov.     

Trees of extremal connectivity index

The connectivity index w_α(G) of a graph G is the sum of the weights (d(u)d(v))^α of all edges uv of G, where α is a real number (α≠0), and d(u) denotes the degree of the vertex u. Let T be a tree with n vertices and k pendant vertices. In this paper, we give sharp lower and upper bounds for w₁(T). Also, for -1?α<0, we give a sharp lower bound and a upper bound for w_α(T).     

A note on lattice chains and Delannoy numbers

Fix nonnegative integers n₁,…,n_d and let L denote the lattice of integer points (a₁,…,a_d)∈Z^d satisfying 0?a_i?n_i for 1?i?d. Let L be partially ordered by the usual dominance ordering. In this paper we offer combinatorial derivations of a number of results concerning chains in L. In particular, the results obtained are established without recourse to generating functions or recurrence relations. We begin with an elementary derivation of the number of chains in L of a given size, from which one can deduce the classical expression for the total number of chains in L. Then we derive a second, alternative, expression for the total number of chains in L when d=2. Setting n₁=n₂ in this expression yields a new proof of a result of Stanley [Enumerative Combinatorics, vol. 2, Cambridge University Press, Cambridge, 1999] relating the total number of chains to the central Delannoy numbers. We also conjecture a generalization of Stanley's result to higher dimensions.     

Über die Wurzelschranke für das Minimalgewicht von codes

Let d be the minimum distance of an (n, k) code $\text{C}$, invariant under an abelian group acting transitively on the basis of the ambient space over a field F with char F × n. Assume that $\text{C}$ contains the repetition code, that dim($\text{C}$ ∏ $\text{C}$^⊥) = k ? 1 and that the supports of the minimal weight vectors of $\text{C}$ form a 2-design. Then d² ? d + 1 ? n with equality if and only if the design is a projective plane of order d ? 1. The case d² ? d + 1 = n can often be excluded with Hall's multiplier theorem on projective planes, a theorem which follows easily from the tools developed in this paper Moreover, if d² ? d + 1 > n and F = GF(2) then (d ? 1)² ? n. Examples are the generalized quadratic residue codes.