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On the minimum eccentric distance sum of bipartite graphs with some given parameters

机译：具有给定参数的二部图的最小偏心距总和

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The eccentric distance sum is a novel graph invariant with vast potential in structure activity/property relationships. This graph invariant displays high discriminating power with respect to both biological activity and physical properties. If G = (V-G, E-G) is a simple connected graph, then the eccentric distance sum (EDS) of G is defined as xi(d)(G) = Sigma(v is an element of VG) epsilon(G)(v)D-G(v), where epsilon(G)(v) is the eccentricity of the vertex v and D-G(v) = Sigma(u is an element of VG) d(G)(u, v) is the sum of all distances from the vertex v. Much extremal work has been done on trees with some given parameters by Yu et al. (2011) [25], Li at al. (2012) [19], and Geng et al. (2013) [6]. It is natural to consider this extremal problem on bipartite graphs with some given parameters. In this paper, sharp lower bound on the EDS in the class of all connected bipartite graphs with a given matching number q is determined, the minimum EDS is realized only by the graph K-q,K-n-q. The extremal graph with the minimum EDS in the class of all the n-vertex connected bipartite graphs of odd diameter is characterized. All the extremal graphs having the minimum EDS in the class of all connected n-vertex bipartite graphs with a given vertex connectivity are identified as well. (C) 2015 Elsevier Inc. All rights reserved.
机译：偏心距总和是一个新颖的图不变式，在结构活动/属性关系中具有巨大的潜力。该图不变式在生物活性和物理性质方面均显示出高鉴别力。如果G =（VG，EG）是一个简单的连通图，则G的偏心距总和（EDS）定义为xi（d）（G）= Sigma（v是VG的元素）epsilon（G）（v ）DG（v），其中epsilon（G）（v）是顶点v的偏心率，而DG（v）= Sigma（u是VG的元素）d（G）（u，v）是所有顶点的和Yu等人已经对具有某些给定参数的树木进行了许多极端工作。 （2011）[25]，李等。 （2012）[19]，和耿等。 （2013）[6]。在带有某些给定参数的二部图上考虑这个极值问题是很自然的。本文确定了具有给定匹配数q的所有连通二部图的类中EDS的尖锐下界，仅通过图K-q，K-n-q实现最小EDS。在具有奇数直径的所有n个顶点相连的二部图的类中，表征了具有最小EDS的极值图。在具有给定顶点连通性的所有已连接n顶点二部图的类别中，也将识别具有最小EDS的所有极值图。 （C）2015 Elsevier Inc.保留所有权利。

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