Spectral Generalizations of Line Graphs

Abstract

Line graphs have the property that their least eigenvalue is greater than or equal to –2, a property shared by generalized line graphs and a finite number of so-called exceptional graphs. This book deals with all these families of graphs in the context of their spectral properties. The authors discuss the three principal techniques that have been employed, namely 'forbidden subgraphs', 'root systems' and 'star complements'. They bring together the major results in the area, including the recent construction of all the maximal exceptional graphs. Technical descriptions of these graphs are included in the appendices, while the bibliography provides over 250 references. This will be an important resource for all researchers with an interest in algebraic graph theory.

Eigenvalue bounds for the signless laplacian

“…Together with some colleagues we continue to consider conjectures presented in this paper. In particular, we expect results related to Conjectures 6,7,22,23 and 24. …”

On independent star sets in finite graphs

“…By [10, Theorem 5.3.1], G has an exceptional star complement H . By [10,Theorem 2.3.20], H has order at most 8, and so t ∈ {7, 8}. We have seen that when t = 7, G is the graph of Example 2.2, and that this graph arises precisely when H is complete.…”

“…Accordingly it suffices to deal with a connected graph G. Since −2 is the least eigenvalue of G, G is either a generalized line graph or an exceptional graph (see [10]). Since −2 is a main eigenvalue, we know that G is not a line graph; in fact, we have:…”

ILIGRA: An Efficient Inverse Line Graph Algorithm

“…Figures 4c, 4d, 4e and 4f illustrate the trends for the algorithms' running times for line graphs with high link density p H = 0.50 and p H = 0.65. Iligra is the fastest algorithm for line graphs with 5 The implementations are available on the authors' web page: http://www.nas.ewi.tudelft.nl/people/Stojan/code/ILIGRA.zip 6 LEDA: http://www.algorithmic-solutions.com/leda/ 7 Intel(R) Core(TM) 2 Duo CPU T9600 on 2 x 2.80GHz; 4GB RAM memory 8 The link density of a given line graph…”