Electronic Notes in Discrete Mathematics最新文献

The total distance or Wiener index $W\left(G\right)$ of a connected graph G is defined as the sum of distances between all pairs of vertices in G. In 1991, Šoltés posed the problem of finding all graphs G such that the equality $W\left(G\right)=W(G-v)$ holds for all their vertices v. Up to now, the only known graph with this property is the cycle C₁₁. Our main object of study is a relaxed version of this problem: Find graphs for which total distance does not change when a particular vertex is removed. We show that there are infinitely many graphs that satisfy this property. This gives hope that Šoltes's problem may have also some solutions distinct from C₁₁.

We introduce a family of Eulerian digraphs, $E$, associated with Dyck words. We provide the algorithms implementing the bijection between $E$ and $W$, the set of Dyck words. To do so, we exploit a binary matrix, that we call Dyck matrix, representing the cycles of an Eulerian digraph.

In this abstract, we construct explicitly, for every k, pairs of non-isomorphic trees with the same restricted U-polynomial; by this we mean that the polynomials agree on terms with degree at most k. The construction is done purely in algebraic terms, after introducing and studying a generalization of the U-polynomial to rooted graphs.

Given two sets of points A and B in a normed plane, we prove that there are two linearly separable sets $A^{\prime }$ and $B^{\prime }$ such that $\mathrm{diam}\left(A^{\prime }\right)\le \mathrm{diam}\left(A\right),\mathrm{diam}\left(B^{\prime }\right)\le \mathrm{diam}\left(B\right)$, and $A^{\prime }\cup B^{\prime }=A\cup B$. As a result, some Euclidean clustering algorithms are adapted to normed planes, for instance, those that minimize the maximum, the sum, or the sum of squares of the diameters (or the radii) of k clusters. Some specific solutions are presented for $k=2$ and $k=3$ that minimize the diameter of the clusters. The 2-clustering problem when two different bounds are imposed to the diameters is also studied.

A partition $P=\{V_1,\dots ,V_m\}$ of the vertex set V of a graph is regular if, for all i, j, the number of neighbors which a vertex in V_i has in the set V_j is independent of the choice of vertex in V_i. The natural generalization of a regular partition, which makes sense also for non-regular graphs, is the so-called weight-regular partition, which gives to each vertex $u\in V$ a weight which equals the corresponding entry ${\nu }_u$ of the Perron eigenvector ν. In this work we investigate when a weight-regular partition of a graph is regular in terms of double stochastic matrices. Inspired by a characterization of regular graphs by Hoffman, we provide a new characterization of weight-regular partitions by using a Hoffman-like polynomial.

In this paper we demonstrate that the result by Zelikovskij concerning Königs problem for abelian permutation groups, reported in a recent survey, is false. We propose in this place two results on 2-closed abelian permutation groups which concern the same topic in a more general setting.

In this paper, we prove that the number of 4-contractible edges (edges that after contraction do not change the connectivity of the initial graph) of a 4-connected graph G is at least $(1/28){\sum }_{x\in V_{\ge 5}\left(G\right)}{\mathrm{deg}}_G\left(x\right)$, where $V_{\ge 5}\left(G\right)$ denotes the set of those vertices of G which have degree greater than or equal to 5.

This is the refinement of the result proved by Ando et al. [On the number of 4-contractible edges in 4-connected graphs, J. Combin. Theory Ser. B 99 (2009) 97–109].

This work focuses on the visibility of exterior of a polygon. We show a lower bound for the Fortress Problem applied to polygons P having n vertices in terms of the number of reflex vertices, the number of convex vertices and the number of pockets that are found when the convex hull is made on P. The results are related to the task of the geometric data acquisition for architectural surveys through techniques such as laser scanner.

A family $A$ of k-subsets of $\{1,2,\dots ,N\}$ is a Sidon system if the sumsets $A+A^{\prime },A,A^{\prime }\in A$ are pairwise distinct. We show that the largest cardinality $F_k\left(N\right)$ of a Sidon system of k-subsets of $\left[N\right]$ satisfies $F_k\left(N\right)\le \left(\begin{array}{c}N-1\\ k-1\end{array}\right)+N-k$ and the asymptotic lower bound $F_k\left(N\right)={\Omega }_k\left(N^{k-1}\right)$. More precise bounds on $F_k\left(N\right)$ are obtained for $k\le 3$. We also obtain the threshold probability for a random system to be Sidon for $k=2$ and 3.

We prove that for pairwise co-prime numbers $k_1,\dots ,k_d\ge 2$ there does not exist any infinite set of positive integers $A$ such that the representation function $r_A\left(n\right)=\#\left\{\right(a_1,\dots ,a_d)\in A^d:k_1{a}_1+\dots +k_d{a}_d=n\}$ becomes constant for n large enough. This result is a particular case of our main theorem, which poses a further step towards answering a question of Sárközy and Sós and widely extends a previous result of Cilleruelo and Rué for bivariate linear forms (Bull. of the London Math. Society 2009).