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Distance spectra and distance energies of iterated line graphs of regular graphs

Ramane H.S. (Department of Mathematics Gogte Institute of Technology Udyambag, Belgaum, India)
Revankar D.S. (Department of Mathematics Gogte Institute of Technology Udyambag, Belgaum, India)
Gutman Ivan (Faculty of Science, Kragujevac)
Walikar H.B. (Department of Computer Science Karnatak University Dharwad, India)

The distance or D-eigenvalues of a graph G are the eigenvalues of its distance matrix. The distance or D-energy ED(G) of the graph G is the sum of the absolute values of its D-eigenvalues. Two graphs G1 and G2 are said to be D-equienergetic if ED(G1) = ED(G2). Let F1 be the 5-vertex path, F2 the graph obtained by identifying one vertex of a triangle with one end vertex of the 3-vertex path, F3 the graph obtained by identifying a vertex of a triangle with a vertex of another triangle and F4 be the graph obtained by identifying one end vertex of a 4-vertex star with a middle vertex of a 3-vertex path. In this paper we show that if G is r-regular, with diam(G)≤ 2, and Fi,i = 1,2,3,4, are not induced subgraphs of G, then the k-th iterated line graph Lk(G) has exactly one positive D-eigenvalue. Further, if G is r-regular, of order n, diam(G)≤2, and G does not have Fi,i=1,2,3,4, as an induced subgraph, then for k ≥1, ED(Lk(G)) depends solely on n and r. This result leads to the construction of non D-cospectral, D-equienergetic graphs having same number of vertices and same number of edges.