In this paper, we propose a new edge-based matching kernel for graphs by using discrete-time quantum walks. To this end, we commence by transforming a graph into a directed line graph. The reasons of using the line graph structure are twofold. First, for a graph, its directed line graph is a dual representation and each vertex of the line graph represents a corresponding edge in the original gr...

The line graph of a graph with signed edges carries vertex signs. A vertex-signed graph is consistent if every circle (cycle, circuit) has positive vertex-sign product. Acharya, Acharya, and Sinha recently characterized line-consistent signed graphs, i.e., edge-signed graphs whose line graphs, with the naturally induced vertex signature, are consistent. Their proof applies Hoede’s relatively di...

Trapezoid graphs are intersection graphs of trapezoids between two horizontal lines. They generalize both interval graphs and permutation graphs. Interval graphs are intersection graphs of intervals on the real line. Permutation graphs are intersection graphs of straight lines between two horizontal lines. They can be represented by a permutation diagram consisting of a pair of horizontal lines...

This is the first half of a two-part paper devoted to on-line 3-colorable graphs. Here on-line 3-colorable triangle-free graphs are characterized by a finite list of forbidden induced subgraphs. The key role in our approach is played by the family of graphs which are both triangleand (2K2 + K1)-free. Characterization of this family is given by introducing a bipartite modular decomposition conce...

We study on-line colorings of certain graphs given as intersection graphs of objects “between two lines”, i.e., there is a pair of horizontal lines such that each object of the representation is a connected set contained in the strip between the lines and touches both. Some of the graph classes admitting such a representation are permutation graphs (segments), interval graphs (axisaligned recta...

In this paper, we give infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval [−1− √ 2,−2) and also infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval [α1,−1− √ 2) where α1 is the smallest root(≈ −2.4812) of the polynomial x3 + 2x2 − 2x − 2. From these results, we ...

iv Acknowledgements v List of Tables vii List of Graphs vii

Graphs with circular symmetry, called webs, are relevant w.r.t. describing the stable set polytopes of two larger graph classes, quasi-line graphs [8,12] and claw-free graphs [7,8]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem [9]. Ben Rebea conjectured a description for quasi-line graphs, see [12]; Chudnovsky and Seymour [2] v...

A graph G is a quasi-line graph if for every vertex v ∈ V (G), the set of neighbors of v in G 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. Hadwiger’s conjecture states that if a graph G is not t-colorable then it contains Kt+1 as a minor. This conjecture has been proved for line graphs by Reed and Seymour [10]. We...

Reed’s ω, ∆, χ conjecture proposes that every graph satisfies χ ≤ d 12 (∆ + 1 + ω)e; it is known to hold for all claw-free graphs. In this paper we consider a local strengthening of this conjecture. We prove the local strengthening for line graphs, then note that previous results immediately tell us that the local strengthening holds for all quasi-line graphs. Our proofs lead to polytime algori...