Equivalence of four descriptions of generalized line graphs

Abstract

(1984), 182-192]. This process includes a short method of finding the aforementioned graphs, analogous to the method found by Beineke [J Combin Theory 9 (1970), 129-135] to obtain all minimal nonline graphs. ᭧

Graphs with least eigenvalue −2: Ten years on

“…The interested reader is referred to earlier papers by Torgašev [157,158], and to those of Vijayakumar et al [118,119,120,140,159], some of which are recent.…”

Characterizations of the family of all generalized line graphs-finite and infinite- and classification of the family of all graphs whose least eigenvalues \ge - 2

“…If u is a vertex of two members of F, say F α , F β , then let ξ(u) = α + β. Suppose that u belongs to exactly one member, say [4,5] and five different methods of computing M f have been found in [4,9,13,5,14]. Countably infinite graphs in L also have been studied: in [10], it has been shown that any countably infinite connected graph with least eigenvalue −2 is a generalized line graph and in [11], all countably infinite connected graphs with least eigenvalues > −2 have been determined.…”

“…Thus it follows that M f and M are same. In [14], M f has been computed by using the finite version of Theorem 10 but not directly. (See [4,9,5] for different methods.)…”

“…• For each i ∈ {12, 13,14,15,16,17,18,19,20,21,22,25,26,28, 29, 30}, (E3) does not hold in G i . • For each i ∈ {4, 7, 8, 9}, (E4) does not hold in G i .…”

“…Using Theorem 2 effectively, the set of all minimal forbidden graphs for the family of all line graphs has been found in [1]. In this article, the next result-its finite version has been derived in [14]-is similarly used for determining the set of all minimal forbidden graphs for the family of all generalized line graphs.…”