In this paper, we introduce a new multivalued function ~ called the line hypergraph. The function ~ generalizes two classical concepts at once, namely, of the line graph and the dual hypergraph. In terms of this function, proofs of some known theorems on line graphs can be unified and their more general versions can be obtained. Three such theorems are considered here, namely, the Berge theorem...

Statistical graphs—in particular, line graphs and bar graphs—are efficient means of communication in a wide range of non-expert settings. In communication settings, statistical graphs do not only serve as visualizations of individual data points but also provide visual access to various aspects of the information contained in data. Moreover, specific types of graphs are better means for providi...

We determine the exact treewidth of the line graph of the complete graph. More generally, we determine the exact treewidth of the line graph of a regular complete multipartite graph. For an arbitrary complete multipartite graph, we determine the treewidth of the line graph up to a lower order term.

Motivated by investigations of rainbow matchings in edge colored graphs, we introduce the notion of color-line graphs that generalizes the classical concept of line graphs in a natural way. Let H be a (properly) edge-colored graph. The (proper) color-line graph CL(H) of H has edges of H as vertices, and two edges of H are adjacent in CL(H) if they are incident in H or have the same color. We gi...

A graph G is a quasi-line graph if for every vertex v, the set of neighbors of v can be expressed as the union of two cliques. The class of quasi-line graphs is a proper superset of the class of line graphs. A theorem of Shannon’s implies that if G is a line graph then it can be properly colored using no more than 32ω(G) colors, where ω(G) is the size of the largest clique in G. In this paper w...

Thomassen conjectured [8] that every 4-connected line graph is hamiltonian. An hourglass is a graph isomorphic to K5−E(C), where C is a cycle of length 4 in K5. In [2], it is shown that every 4-connected line graph without an induced subgraph isomorphic to the hourglass is hamiltonian connected. In this note, we prove that every 3-connected, essentially 4-connected hourglass-free line graph is ...

Boxicity of a graph H , denoted by box(H), is the minimum integer k such that H is an intersection graph of axis-parallel kdimensional boxes in R. In this paper, we show that for a line graph G of a multigraph, box(G) ≤ 2∆(⌈log 2 log 2 ∆⌉ + 3) + 1, where ∆ denotes the maximum degree of G. Since ∆ ≤ 2(χ−1), for any line graph G with chromatic number χ, box(G) = O(χ log 2 log 2 (χ)). For the d-di...

The line graph of G, denoted L(G), is the graph with vertex set E(G), where vertices x and y are adjacent in L(G) iff edges x and y share a common vertex in G. In this paper, we determine all graphs G for which L(G) is a circulant graph. We will prove that if L(G) is a circulant, then G must be one of three graphs: the complete graph K4, the cycle Cn, or the complete bipartite graph Ka,b, for s...

In this note, we give expressions for the first(second) Zagreb coindex, second Zagreb index(coindex), third Zagreb index and first hyper-Zagreb index of the line graphs of subdivision graphs of 2D-lattice, nanotube and nanotorus of TUC4C8[p, q] and obtain upper bounds for Wiener index and degree-distance index of these graphs. This note continue the program of computing certain topological indi...

We classify the distance-regular Cayley graphs with least eigenvalue −2 and diameter at most three. Besides sporadic examples, these comprise of the lattice graphs, certain triangular graphs, and line graphs of incidence graphs of certain projective planes. In addition, we classify the possible connection sets for the lattice graphs and obtain some results on the structure of distance-regular C...