Abstract
The distance distribution (dd) of a connected graph of diameter k is (D1,D2,...,Dk), where Di is the number of pairs of vertices at distance i from one another. The common neighbor distribution (nd) is (n0,n1,n2,...,nn–2), where ni is the number of pairs of vertices having i common neighbors. These and other sequences have been introduced recently as tools in distinguishing pairs of nonisomorphic graphs (dd(G) works best for graphs of large diameter; whereas, nd(G) is more useful for graphs of small diameter). They have also been used to study structural similarity in graphs sharing a common sequence.
If G has large diameter, its complement 0000123 has small diameter. In this paper, we use the concept of dominating sets to characterize graphs for which dd(G)=dd(0000123), and graphs for which nd(G)=nd(0000123). In the final section, we introduce a new graphical distribution being studied by Capobianco. He defines the geodesic distribution (gd) of a connected graph as (g1,g2,g3,...), where gi is the number of pairs of vertices having i shortest paths (geodesics) between them. We discuss gd(G) and examine its relationships to dd(G) and nd(G).
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© 1984 Springer-Verlag
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Buckley, F. (1984). Equalities involving certain graphical distributions. In: Koh, K.M., Yap, H.P. (eds) Graph Theory Singapore 1983. Lecture Notes in Mathematics, vol 1073. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0073116
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DOI: https://doi.org/10.1007/BFb0073116
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