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Using Minimum Degree to Bound Average Distance
, 1998
"... We show the average distance of a connected graph with n vertices, e edges and minimum degree d satisfies # j (n+1)n(n1)2e d+1 k n (n  1) # n + 1 d + 1  2e (d + 1) n (n  1) . This improves conjectures of the computer program Gra#ti [8], and He and Li [11] and the results of Kouider ..."
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We show the average distance of a connected graph with n vertices, e edges and minimum degree d satisfies # j (n+1)n(n1)2e d+1 k n (n  1) # n + 1 d + 1  2e (d + 1) n (n  1) . This improves conjectures of the computer program Gra#ti [8], and He and Li [11] and the results of Kouider and Winkler [13]. The bound is sharp since complete graphs yield equality. 1 Introduction Let G = (V, E) be a connected, simple, undirected graph on n vertices. For each unordered pair of vertices {u, v} in G, select a path of minimal length. Designate the corresponding directed paths from u to v and from v to u as, P (u, v) and P (v, u) , respectively. The number of edges in P (u, v) is the distance # (u, v) from u to v and the average distance of G is = 1 n (n  1) X u,v #V # (u, v) . Total and average distance are not only interesting invariants of graphs in their own right but are also used for studying properties or classifying graphical systems that depend on the number of edg...
Lists of FaceRegular Polyhedra
, 1999
"... We introduce a new notion that connects the combinatorial concept of regularity with the geometrical notion of face transitivity. This new notion implies finiteness results in the case of bounded maximal face size. We give lists of structures for some classes and investigate polyhedra with constant ..."
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We introduce a new notion that connects the combinatorial concept of regularity with the geometrical notion of face transitivity. This new notion implies finiteness results in the case of bounded maximal face size. We give lists of structures for some classes and investigate polyhedra with constant vertex degrees and faces of only two sizes.
Fullerenes and Coordination Polyhedra versus HalfCubes Embeddings
, 1997
"... A fullerene F n is a 3regular (or cubic) polyhedral carbon molecule for which the n vertices  the carbons atoms  are arranged in 12 pentagons and ( n 2 \Gamma 10) hexagons. Only a finite number of fullerenes are expected to be, up to scale, isometrically embeddable into a hypercube. Looking fo ..."
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A fullerene F n is a 3regular (or cubic) polyhedral carbon molecule for which the n vertices  the carbons atoms  are arranged in 12 pentagons and ( n 2 \Gamma 10) hexagons. Only a finite number of fullerenes are expected to be, up to scale, isometrically embeddable into a hypercube. Looking for the list of such fullerenes, we first check the embeddability of all fullerenes F n for n ! 60 and of all preferable fullerenes C n for n ! 86 and their duals. Then, we consider some infinite families, including fullerenes with icosahedral symmetry, which describe virus capsids, onionlike metallic clusters and geodesic domes. Quasiembeddings and fullerene analogues are considered. We also present some results on chemically relevant polyhedra such as coordination polyhedra and cluster polyhedra. Finally we conjecture that the list of known embeddable fullerenes is complete and present its relevance to the Katsura model for vesicles cells. Contents 1 Introduction and Basic Properties 2 1...
Characterization of Isospectral Graphs Using Graph Invariants and Derived Orthogonal Parameters
 J. Chem. Inf. Comput. Sci. 1998
"... from TIs have been used in discriminating a set of isospectral graphs. Results show that lower order connectivity and information theoretic TIs suffer from a high degree of redundancy, whereas higher order indices can characterize the graphs reasonably well. On the other hand, PCs derived from the T ..."
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from TIs have been used in discriminating a set of isospectral graphs. Results show that lower order connectivity and information theoretic TIs suffer from a high degree of redundancy, whereas higher order indices can characterize the graphs reasonably well. On the other hand, PCs derived from the TIs had no redundancy for the set of isospectral graphs studied. 1.