We determine the forbidden induced subgraphs for the intersection of the classes of chordal bipartite graphs and line graphs of acyclic directed graphs. This is a first step towards finding the forbidden induced subgraphs for the class of line graphs of directed graphs.

The diameter of a connected graph G, denoted by diam (G), is the maximum distance between any pair of vertices of G. Let L (G) be the line graph of G. We establish necessary and sufficient conditions under which for a given integer k³2, diam (L(G))£k.

A signed graph (or, in short, sigraph) S = (Su, s) consists of an underlying graph Su:= G = (V, E) and a function s: E (Su) ®{+, -}, called the signature of S. A marking of S is a function m: V (S) ®{+, -}. The canonical marking of a signed graph S, denoted ms, is given asms (v):= Õ vw2E(S) s (vw).The line graph of a graph G, denoted L (G), is the graph in which edges of G are represented as vertices, two of these vertices are adjacent if the corresponding edges are adjacent in G. There are three notions of a line signed graph of a signed graph S = (Su, s) in the literature, viz., L (S), Lx (S) and L· (S), all of which have L (Su) as their underlying graph, only the rule to assign signs to the edges of L (Su) differ. Every edge ee ′in L (S) is negative whenever both the adjacent edges e and e’ in S are negative, an edge ee′in Lx (S) has the product s (e) s (e′) as its sign and an edge ee′in L· (S) has ms (v) as its sign, where vÎ V (S) is a common vertex of edges e and e′.The line-cut graph (or, in short, lict graph) of a graph G = (V,E), denoted by Lc (G), is the graph with vertex set E (G) È C (G), where C (G) is the set of cut-vertices of G, in which two vertices are adjacent if and only if they correspond to adjacent edges of G or one vertex corresponds to an edge e of G and the other vertex corresponds to a cut-vertex c of G such that e is incident with c.In this paper, we introduce dot-lict signed graph (or ·-lict signed graph) L·c (S), which has Lc (Su) as its underlying graph. Every edge uv in L·c (S) has the sign ms (p), if u, v Î E (S) and p Î V (S) is a common vertex of these edges, and it has the sign ms (v), if u Î E (S) and v Î C (S). We characterize signed graphs on Kp, p³2, on cycle Cn and on Km, n which are ·-lict signed graphs or ·-line signed graphs, characterize signed graphs S so that L·c (S) and L· (S) are balanced. We also establish the characterization of signed graphs S for which S ~ L·c (S), S ~ L· (S), h (S) ~ L·c (S) and h (S) ~ L· (S), here h (S) is negation of S and ~ stands for switching equivalence.

The D􀀀 eigenvalues f 1;  2; : : :;  pg of a graph G are the eigenvalues of its distance matrix D and form its D􀀀 spectrum. The D􀀀 energy, ED(G) of G is given by ED (G) = Σ p i=1 j ij. Two non cospectral graphs with respect to D are said to be D􀀀 equi energetic if they have the same D􀀀 energy. In this paper we show that if G is an r􀀀 regular graph on p vertices with 2r  p 􀀀 1, then the complements of iterated line graphs of G are of diameter 2 and that ED(Lk(G)); k  2 depends only on p and r. This result leads to the construction of regular D􀀀 equi energetic pair of graphs.