Construction And Spectra Of Non-Regular Minimal Graphs
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The number of distinct eigenvalues of the adjacency matrix of graph G is bounded below by d(G)+1, where d is the diameter of the graph. Graphs attaining this lower bound are known as minimal graphs. The spectrum of graph G, where G is a simple and undirected graph is the collection of different eigenvalues of the adjacency matrix with their multiplicities. This paper deals with the construction of non-regular minimal graphs, together with the study of their characteristic polynomial and spectra.Keywords:
Adjacency matrix
Adjacency list
The number of distinct eigenvalues of the adjacency matrix of graph G is bounded below by d(G)+1, where d is the diameter of the graph. Graphs attaining this lower bound are known as minimal graphs. The spectrum of graph G, where G is a simple and undirected graph is the collection of different eigenvalues of the adjacency matrix with their multiplicities. This paper deals with the construction of non-regular minimal graphs, together with the study of their characteristic polynomial and spectra.
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A graph in a certain graph class is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum among all graphs in that class.Bell et al. have identified a subclass within the connected graphs of order n and size m in which minimizing graphs belong (the complements of such graphs are either disconnected or contain a clique of size n 2 ).In this paper we discuss the minimizing graphs of a special class of graphs of order n whose complements are connected and contains exactly one cycle (namely the class U c n of graphs whose complements are unicyclic), and characterize the unique minimizing graph in U c n when n ≥ 20.
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Recently, Milani\v{c} and Trotignon introduced the class of equistarable graphs as graphs without isolated vertices admitting positive weights on the edges such that a subset of edges is of total weight $1$ if and only if it forms a maximal star. Based on equistarable graphs, counterexamples to three conjectures on equistable graphs were constructed, in particular to Orlin's conjecture, which states that every equistable graph is a general partition graph. In this paper we characterize equistarable bipartite graphs. We show that a bipartite graph is equistarable if and only if every $2$-matching of the graph extends to a matching covering all vertices of degree at least $2$. As a consequence of this result, we obtain that Orlin's conjecture holds within the class of complements of line graphs of bipartite graphs. We also connect equistarable graphs to the triangle condition, a combinatorial condition known to be necessary (but in general not sufficient) for equistability. We show that the triangle condition implies general partitionability for complements of line graphs of forests, and construct an infinite family of triangle non-equistable graphs within the class of complements of line graphs of bipartite graphs.
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Trapezoid graphs are the intersection graphs of trapezoids, where every trapezoid has a pair of opposite sides lying on two parallel lines L_{1} and L_{2} of the plane. Strictly between permutation and trapezoid graphs lie the simple-triangle graphs -- also known as PI graphs (for Point-Interval) -- where the objects are triangles with one point of the triangle on L_1 and the other two points (i.e. interval) of the triangle on L_2, and the triangle graphs -- also known as PI^* graphs -- where again the objects are triangles, but now there is no restriction on which line contains one point of the triangle and which line contains the other two. The complexity status of both triangle and simple-triangle recognition problems (namely, the problems of deciding whether a given graph is a triangle or a simple-triangle graph, respectively) have been the most fundamental open problems on these classes of graphs since their introduction two decades ago. Moreover, since triangle and simple-triangle graphs lie naturally between permutation and trapezoid graphs, and since they share a very similar structure with them, it was expected that the recognition of triangle and simple-triangle graphs is polynomial, as it is also the case for permutation and trapezoid graphs. In this article we surprisingly prove that the recognition of triangle graphs is NP-complete, even in the case where the input graph is known to be a trapezoid graph.
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In a graph coloring, each color class induces a disjoint union of isolated vertices. A graph subcoloring generalizes this concept, since here each color class induces a disjoint union of complete graphs. Erdos and, independently, Albertson et al., proved that every graph of maximum degree at most 3 has a 2-subcoloring. We point out that this fact is best possible with respect to degree constraints by showing that the problem of recognizing 2-subcolorable graphs with maximum degree 4 is NP-complete, even when restricted to triangle-free planar graphs. Moreover, in general, for fixed k, recognizing k-subcolorable graphs is NP-complete on graphs with maximum degree at most k2 . In contrast, we show that, for arbitrary k, k-SUBCOLORABILITY can be decided in linear time on graphs with bounded treewidth and on graphs with bounded cliquewidth (including cographs as a specific case).
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Recently, Milanic and Trotignon introduced the class of equistarable graphs as graphs without isolated vertices admitting positive weights on the edges such that a subset of edges is of total weight $1$ if and only if it forms a maximal star. Based on equistarable graphs, counterexamples to three conjectures on equistable graphs were constructed, in particular to Orlin's conjecture, which states that every equistable graph is a general partition graph.
In this paper we characterize equistarable bipartite graphs. We show that a bipartite graph is equistarable if and only if every $2$-matching of the graph extends to a matching covering all vertices of degree at least $2$. As a consequence of this result, we obtain that Orlin's conjecture holds within the class of complements of line graphs of bipartite graphs.
We also connect equistarable graphs to the triangle condition, a combinatorial condition known to be necessary (but in general not sufficient) for equistability. We show that the triangle condition implies general partitionability for complements of line graphs of forests, and construct an infinite family of triangle non-equistable graphs within the class of complements of line graphs of bipartite graphs.
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