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

Previous Article

Next Article

Contents of this Issue

Other Issues


ELibM Journals

ELibM Home

EMIS Home


Pick a mirror

 

DISTANCE SPECTRA AND DISTANCE ENERGIES OF ITERATED LINE GRAPHS OF REGULAR GRAPHS

H. S. Ramane, D. S. Revankar, I. Gutman, and H. B. Walikar

Department 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 India

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 $D$-equienergetic if $E_D(G_1)=E_D(G_2)$. Let $F_1$ be the 5-vertex path, $F_2$ the graph obtained by identifying one vertex of a triangle with one end vertex of the 3-vertex 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 4-vertex star with a middle vertex of a 3-vertex path. In this paper we show that if $G$ is $r$-regular, with $\diam(G)\leq2$, 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)


Electronic fulltext finalized on: 23 Apr 2009. This page was last modified: 22 Oct 2009.

© 2009 Mathematical Institute of the Serbian Academy of Science and Arts
© 2009 ELibM and FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition