Abstract. The distance or D-eigenvalues of a graph G are the eigenvalues of its distance matrix. The distance or D-energy E D (G) of the graph G is the sum of the absolute values of its D-eigenvalues. Two graphs G 1 and G 2 are said to be
The center C(G) and the periphery P (G) of a connected graph G consist of the vertices of minimum and maximum eccentricity, respectively. Almostperipheral (AP) graphs are introduced as graphs G with |P (G)| = |V (G)|−1 (and |C(G)| = 1). AP graph of radius r is called an r-AP graph. Several constructions of AP graph are given, in particular implying that for any r ≥ 1, any graph can be embedded as an induced subgraph into some r-AP graph. A decomposition of AP-graphs that contain cut-vertices is presented. The r-embedding index Φ r (G) of a graph G is introduced as the minimum number of vertices which have to be added to G such that the obtained graph is an r-AP graph. It is proved that Φ 2 (G) ≤ 5 holds for any non-trivial graphs and that equality holds if and only if G is a complete graph.
If G is a molecular graph with n vertices and if ?1, ?2, ..., ?n are its
eigenvalues, then the energy of G is equal to E(G) = |?1| + |?2|+ ... +
|?n|. If E(G) > 2n - 2, then G is said to be hyperenergetic. We show that no
H?ckel graph (= the graph representation of a conjugated hydrocarbon within
the H?ckel molecular orbital model) is hyperenergetic.
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