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1.
A graph is called supermagic if it admits a labelling of the edges by pairwise different consecutive positive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. A graph G is called conservative if it admits an orientation and a labelling of the edges by integers {1,…,|E(G)|} such that at each vertex the sum of the labels on the incoming edges is equal to the sum of the labels on the outgoing edges. In this paper we deal with conservative graphs and their connection with the supermagic graphs. We introduce a new method to construct supermagic graphs using conservative graphs. Inter alia we show that the union of some circulant graphs and regular complete multipartite graphs are supermagic. 相似文献
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Andrea Semani?ová 《Discrete Mathematics》2006,306(18):2263-2269
A graph is called magic (supermagic) if it admits a labelling of the edges by pairwise different (and consecutive) integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. In the paper, we characterize magic circulant graphs and 3-regular supermagic circulant graphs. We establish some conditions for supermagic circulant graphs. 相似文献
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We show that regular median graphs of linear growth are the Cartesian product of finite hypercubes with the two-way infinite path. Such graphs are Cayley graphs and have only two ends.For cubic median graphs G the condition of linear growth can be weakened to the condition that G has two ends. For higher degree the relaxation to two-ended graphs is not possible, which we demonstrate by an example of a median graph of degree four that has two ends, but nonlinear growth. 相似文献
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Let Γ be a regular graph with n vertices, diameter D, and d + 1 different eigenvalues λ > λ1 > ··· > λd. In a previous paper, the authors showed that if P(λ) > n − 1, then D ≤ d − 1, where P is the polynomial of degree d − 1 which takes alternating values ± 1 at λ1, …, λd. The graphs satisfying P(λ) = n − 1, called boundary graphs, have shown to deserve some attention because of their rich structure. This paper is devoted to the study of this case and, as a main result, it is shown that those extremal (D = d) boundary graphs where each vertex have maximum eccentricity are, in fact, 2-antipodal distance-regular graphs. The study is carried out by using a new sequence of orthogonal polynomials, whose special properties are shown to be induced by their intrinsic symmetry. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 123–140, 1998 相似文献
5.
An induced matching of a graph G is a matching having no two edges joined by an edge. An efficient edge dominating set of G is an induced matching M such that every other edge of G is adjacent to some edge in M. We relate maximum induced matchings and efficient edge dominating sets, showing that efficient edge dominating sets are maximum induced matchings, and that maximum induced matchings on regular graphs with efficient edge dominating sets are efficient edge dominating sets. A necessary condition for the existence of efficient edge dominating sets in terms of spectra of graphs is established. We also prove that, for arbitrary fixed p≥3, deciding on the existence of efficient edge dominating sets on p-regular graphs is NP-complete. 相似文献
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Flavia Bonomo Guillermo Durán Martín D. Safe Annegret K. Wagler 《Discrete Applied Mathematics》2013,161(13-14):1925-1942
A graph is balanced if its clique-matrix contains no edge–vertex incidence matrix of an odd chordless cycle as a submatrix. While a forbidden induced subgraph characterization of balanced graphs is known, there is no such characterization by minimal forbidden induced subgraphs. In this work, we provide minimal forbidden induced subgraph characterizations of balanced graphs restricted to graphs that belong to one of the following graph classes: complements of bipartite graphs, line graphs of multigraphs, and complements of line graphs of multigraphs. These characterizations lead to linear-time recognition algorithms for balanced graphs within the same three graph classes. 相似文献
8.
Dong Ye 《Discrete Mathematics》2018,341(5):1195-1198
It was conjectured by Mkrtchyan, Petrosyan and Vardanyan that every graph with has a maximum matching such that any two -unsaturated vertices do not share a neighbor. The results obtained in Mkrtchyan et al. (2010), Petrosyan (2014) and Picouleau (2010) leave the conjecture unknown only for -regular graphs with . All counterexamples for -regular graphs given in Petrosyan (2014) have multiple edges. In this paper, we confirm the conjecture for all -regular simple graphs and also -regular multigraphs with . 相似文献
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A graph G is said to be retarded regular if there is a positive integral number s such that the number of walks of length s starting at vertices of G is a constant function. Regular and semiregular graphs are retarded regular with s?=?1 and s\!≤ \!2, respectively. We prove that any retarded regular connected graph is either regular or semiregular. 相似文献
11.
Nick C. Fiala 《Discrete Mathematics》2006,306(23):3083-3096
In this paper, we begin the determination of all primitive strongly regular graphs with chromatic number equal to 5. Using eigenvalue techniques, we show that there are at most 43 possible parameter sets for such a graph. For each parameter set, we must decide which strongly regular graphs, if any, possessing the set are 5-chromatic. In this way, we deal completely with 34 of these parameter sets using eigenvalue techniques and computer enumerations. 相似文献
12.
Given a graph G, the modularity of a partition of the vertex set measures the extent to which edge density is higher within parts than between parts; and the modularity of G is the maximum modularity of a partition.We give an upper bound on the modularity of r-regular graphs as a function of the edge expansion (or isoperimetric number) under the restriction that each part in our partition has a sub-linear numbers of vertices. This leads to results for random r-regular graphs. In particular we show the modularity of a random cubic graph partitioned into sub-linear parts is almost surely in the interval (0.66, 0.88).The modularity of a complete rectangular section of the integer lattice in a fixed dimension was estimated in Guimer et. al. [R. Guimerà, M. Sales-Pardo and L.A. Amaral, Modularity from fluctuations in random graphs and complex networks, Phys. Rev. E 70 (2) (2004) 025101]. We extend this result to any subgraph of such a lattice, and indeed to more general graphs. 相似文献
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We prove that there is an absolute constant C>0 so that for every natural n there exists a triangle‐free regular graph with no independent set of size at least \({{C}}\sqrt{{{n}}\log{{n}}}\). © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 244–249, 2010 相似文献
14.
We describe coedge regular graphs such that antineighborhoods of their vertices are coedge regular graphs with the same value
of the parameterμ. As a consequence of the main theorem, we obtain a classification of coedge regular graphs without 3-stars.
Translated fromMatematicheskie Zametki, Vol. 60, No. 4, pp. 495–503, October, 1996.
This research was supported by the Russion Foundation for Basic Research under grants No. 93-01-01529 and No. 94-01-00802a. 相似文献
15.
It is well known that any finite simple graph Γ is an induced subgraph of some exponentially larger strongly regular graph Γ (e.g., [2, 8]). No general polynomial‐size construction has been known. For a given finite simple graph Γ on υ vertices, we present a construction of a strongly regular graph Γ on O(υ4) vertices that contains Γ as its induced subgraph. A discussion is included of the size of the smallest possible strongly regular graph with this property. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 1–8, 2000 相似文献
16.
M. Abreu 《Discrete Mathematics》2008,308(10):1810-1815
Murty [A generalization of the Hoffman-Singleton graph, Ars Combin. 7 (1979) 191-193.] constructed a family of (pm+2)-regular graphs of girth five and order 2p2m, where p?5 is a prime, which includes the Hoffman-Singleton graph [A.J. Hoffman, R.R. Singleton, On Moore graphs with diameters 2 and 3, IBM J. (1960) 497-504]. This construction gives an upper bound for the least number f(k) of vertices of a k-regular graph with girth 5. In this paper, we extend the Murty construction to k-regular graphs with girth 5, for each k. In particular, we obtain new upper bounds for f(k), k?16. 相似文献
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The main aim of this short paper is to answer the following question. Given a fixed graph H, for which values of the degree d does a random d-regular graph on n vertices contain a copy of H with probability close to one? 相似文献
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An (r, n)-split coloring of a complete graph is an edge coloring with r colors under which the vertex set is partitionable into r parts so that for each i, part i does not contain Kn in color i. This generalizes the notion of split graphs which correspond to (2, 2)-split colorings. The smallest N for which the complete graph KN has a coloring which is not (r, n)-split is denoted by ƒr(n). Balanced (r,n)-colorings are defined as edge r-colorings of KN such that every subset of [N/r] vertices contains a monochromatic Kn in all colors. Then gr(n) is defined as the smallest N such that KN has a balanced (r, n)-coloring. The definitions imply that fr(n) gr(n). The paper gives estimates and exact values of these functions for various choices of parameters. 相似文献