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.

In $2011$, Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. Since then peg solitaire and related games have been considered on many graph classes. One of the main goals is the characterization of solvable graphs. To this end, different graph operations, such as joins and Cartesian products, have been considered in the past. In this article, we continue this venue of research by investigating line graphs. Instead of playing peg solitaire on the line graph $L(G)$ of a graph $G$, we introduce a related game called stick solitaire and play it on $G$. This game is examined on several well-known graph classes, for example complete graphs and windmills. In particular, we prove that most of them are stick-solvable. We also present a family of graphs which contains unsolvable graphs in stick solitaire. Naturally, the Fool's stick solitaire number is an object of interest, which we compute for the previously considered graph classes.

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.

Let n  1 be an integer. The hypercube Qn is the graph whose vertex set is f0; 1gn, where two n-tuples are adjacent if they differ in precisely one coordinate. This graph has many applications in Computer sciences and other area of sciences. In the graph Qn, the layer Lk is the set of vertices with exactly k 1’ s, namely, vertices of weight k, 1  k  n. The hyper-star graph B(n; k) is the subgraph of Qn induced by layers Lk and Lk+1; 0 < k < n. In this paper, we determine the spectrum of the hyperstar graph B(n; k) and L(B(n; k)), where L(B(n; k)) is the line graph of the graph B(n; k). In particular, we show that the graph L(B(n; k)) is an integral graph, that is, all of its eigenvalues are integers. In this paper, we investigate some of the algebraic properties of the graph B(n; k) and its line graph L(B(n; k)). In particular, we determine the spectrum of these graphs.

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.