Nonplanarity of Iterated Line Graphs

The 1-crossing index of a graph G is the smallest integer k such that the k th iterated line graph of G has crossing number greater than 1. In this paper, we show that the 1-crossing index of a graph is either infinite or it is at most 5. Moreover, we give a full characterization of all graphs with respect to their 1-crossing index.

Trees T satisfying W(L3(T))= W(T)

Let G be a graph. Denote by Li(G) its i-iterated line graph and denote by W(G) its Wiener index. We find an infinite class of trees T satisfying W(L3(T)) = W(T), which disproves a conjecture of Dobrynin and Entringer [Electronic Notes in Discrete Math. 22 (2005) 469-475].

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

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.

Maximum Degree Growth of the Iterated Line Graph

Let $\Delta_k$ denote the maximum degree of the $k^{\rm th}$ iterated line graph $L^k(G)$. For any connected graph $G$ that is not a path, the inequality $\Delta_{k+1}\leq 2\Delta_k-2$ holds. Niepel, Knor, and Šoltés have conjectured that there exists an integer $K$ such that, for all $k\geq K$, equality holds; that is, the maximum degree $\Delta_k$ attains the greatest possible growth. We prove this conjecture using induced subgraphs of maximum degree vertices and locally maximum vertices.