755 research outputs found

    On the extremal properties of the average eccentricity

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    The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity ecc(G)ecc (G) of a graph GG is the mean value of eccentricities of all vertices of GG. The average eccentricity is deeply connected with a topological descriptor called the eccentric connectivity index, defined as a sum of products of vertex degrees and eccentricities. In this paper we analyze extremal properties of the average eccentricity, introducing two graph transformations that increase or decrease ecc(G)ecc (G). Furthermore, we resolve four conjectures, obtained by the system AutoGraphiX, about the average eccentricity and other graph parameters (the clique number, the Randi\' c index and the independence number), refute one AutoGraphiX conjecture about the average eccentricity and the minimum vertex degree and correct one AutoGraphiX conjecture about the domination number.Comment: 15 pages, 3 figure

    The Randic index and the diameter of graphs

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    The {\it Randi\'c index} R(G)R(G) of a graph GG is defined as the sum of 1/\sqrt{d_ud_v} over all edges uvuv of GG, where dud_u and dvd_v are the degrees of vertices uu and v,v, respectively. Let D(G)D(G) be the diameter of GG when GG is connected. Aouchiche-Hansen-Zheng conjectured that among all connected graphs GG on nn vertices the path PnP_n achieves the minimum values for both R(G)/D(G)R(G)/D(G) and R(G)−D(G)R(G)- D(G). We prove this conjecture completely. In fact, we prove a stronger theorem: If GG is a connected graph, then R(G)−(1/2)D(G)≥2−1R(G)-(1/2)D(G)\geq \sqrt{2}-1, with equality if and only if GG is a path with at least three vertices.Comment: 17 pages, accepted by Discrete Mathematic

    Eccentric connectivity index

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    The eccentric connectivity index ξc\xi^c is a novel distance--based molecular structure descriptor that was recently used for mathematical modeling of biological activities of diverse nature. It is defined as ξc(G)=∑v∈V(G)deg(v)⋅ϵ(v)\xi^c (G) = \sum_{v \in V (G)} deg (v) \cdot \epsilon (v)\,, where deg(v)deg (v) and ϵ(v)\epsilon (v) denote the vertex degree and eccentricity of vv\,, respectively. We survey some mathematical properties of this index and furthermore support the use of eccentric connectivity index as topological structure descriptor. We present the extremal trees and unicyclic graphs with maximum and minimum eccentric connectivity index subject to the certain graph constraints. Sharp lower and asymptotic upper bound for all graphs are given and various connections with other important graph invariants are established. In addition, we present explicit formulae for the values of eccentric connectivity index for several families of composite graphs and designed a linear algorithm for calculating the eccentric connectivity index of trees. Some open problems and related indices for further study are also listed.Comment: 25 pages, 5 figure

    Problems in extremal graph theory

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    We consider a variety of problems in extremal graph and set theory. The {\em chromatic number} of GG, χ(G)\chi(G), is the smallest integer kk such that GG is kk-colorable. The {\it square} of GG, written G2G^2, is the supergraph of GG in which also vertices within distance 2 of each other in GG are adjacent. A graph HH is a {\it minor} of GG if HH can be obtained from a subgraph of GG by contracting edges. We show that the upper bound for χ(G2)\chi(G^2) conjectured by Wegner (1977) for planar graphs holds when GG is a K4K_4-minor-free graph. We also show that χ(G2)\chi(G^2) is equal to the bound only when G2G^2 contains a complete graph of that order. One of the central problems of extremal hypergraph theory is finding the maximum number of edges in a hypergraph that does not contain a specific forbidden structure. We consider as a forbidden structure a fixed number of members that have empty common intersection as well as small union. We obtain a sharp upper bound on the size of uniform hypergraphs that do not contain this structure, when the number of vertices is sufficiently large. Our result is strong enough to imply the same sharp upper bound for several other interesting forbidden structures such as the so-called strong simplices and clusters. The {\em nn-dimensional hypercube}, QnQ_n, is the graph whose vertex set is {0,1}n\{0,1\}^n and whose edge set consists of the vertex pairs differing in exactly one coordinate. The generalized Tur\'an problem asks for the maximum number of edges in a subgraph of a graph GG that does not contain a forbidden subgraph HH. We consider the Tur\'an problem where GG is QnQ_n and HH is a cycle of length 4k+24k+2 with k≥3k\geq 3. Confirming a conjecture of Erd{\H o}s (1984), we show that the ratio of the size of such a subgraph of QnQ_n over the number of edges of QnQ_n is o(1)o(1), i.e. in the limit this ratio approaches 0 as nn approaches infinity
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