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

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

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

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