Publications de l'Institut Mathématique, Nouvelle Série Vol. 85(99), pp. 39–46 (2009) 

DISTANCE SPECTRA AND DISTANCE ENERGIES OF ITERATED LINE GRAPHS OF REGULAR GRAPHSH. S. Ramane, D. S. Revankar, I. Gutman, and H. B. WalikarDepartment of Mathematics, Gogte Institute of Technology, Udyambag, Belgaum – 590008, India Faculty of Science, University of Kragujevac, P. O. Box 60, 34000 Kragujevac, Serbia Department of Computer Science, Karnatak University, Dharwad – 580003 IndiaAbstract: 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 $D$equienergetic if $E_D(G_1)=E_D(G_2)$. Let $F_1$ be the 5vertex path, $F_2$ the graph obtained by identifying one vertex of a triangle with one end vertex of the 3vertex path, $F_3$ the graph obtained by identifying a vertex of a triangle with a vertex of another triangle and $F_4$ be the graph obtained by identifying one end vertex of a 4vertex star with a middle vertex of a 3vertex path. In this paper we show that if $G$ is $r$regular, with $\diam(G)\leq2$, and $F_i$, $i=1,2,3,4$, are not induced subgraphs of $G$, then the $k$th iterated line graph $L^k(G)$ has exactly one positive $D$eigenvalue. Further, if $G$ is $r$regular, of order $n$, $\diam(G)\leq2$, and $G$ does not have $F_i$, $i=1,2,3,4$, as an induced subgraph, then for $k\geq1$, $E_D(L^k(G))$ 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. Classification (MSC2000): 05C12; 05C50 Full text of the article: (for faster download, first choose a mirror)
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