Maximal reflexive cacti with four cycles: The approach via Smith graphs

Rašajski, Marija; Radosavljević, Zvonko; Mihailovic, Bojana

doi:10.1016/j.laa.2011.04.023

Cited by 10 publications

(9 citation statements)

References 14 publications

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“…In this case maximal means that its supergraphs are not reflexive. It turned out that Smith graphs play an essential role also in the construction of maximal reflexive graphs [7,8,9,11,12] By generalizing the RS-theorem we get a useful instrument for comparing λ 2 with arbitrary a > 0 for many graphs with a cut-vertex.…”

Section: Auxiliary Results and The Rs-theoremmentioning

confidence: 99%

“…Some of the bounds considered so far are: a = [3], a = 1 [4], a = √ 2 [5], a = √ 3 [5]. Reflexive graphs (a = 2) have been investigated in many articles, for example [8,9,10,11,12] where Manuscript received April 11 ; accepted June 30 B. Lj. Mihailović is with the Faculty of Electrical Engineering, University of Belgrade, Serbia, e-mail mihailovicb@etf.rs Theorem 1 of [10] (further, RS-theorem) was often used to prove whether a connected graph with a cut-vertex is reflexive or not.…”

Section: Introductionmentioning

confidence: 99%

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Trees whose second largest eigenvalue does not exceed √ 5+1 / 2

Mihailovic¹

2014

Sci Publ State Univ Novi Pazar Ser A

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Section: Auxiliary Results and The Rs-theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Trees whose second largest eigenvalue does not exceed √ 5+1 / 2

Mihailovic¹

2014

Sci Publ State Univ Novi Pazar Ser A

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“…See, for example, [27] and references therein. Reflexive graphs with a cut vertex are neatly described in [26].…”

Section: Salem Numbers and Salem Graphs: Definitionsmentioning

confidence: 99%

A classification of all 1-Salem graphs

Gumbrell

McKee

2014

LMS J. Comput. Math.

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One way to study certain classes of polynomials is by considering examples that are attached to combinatorial objects. Any graph G has an associated reciprocal polynomial RG, and with two particular classes of reciprocal polynomials in mind one can ask the questions: (a) when is RG a product of cyclotomic polynomials (giving the cyclotomic graphs)? (b) when does RG have the minimal polynomial of a Salem number as its only non-cyclotomic factor (the non-trivial Salem graphs)? Cyclotomic graphs were classified by Smith (Combinatorial structures and their applications, Proceedings of Calgary International Conference, Calgary, AB, 1969 (eds R. Guy, H. Hanani, H. Saver and J. Schönheim; Gordon and Breach, New York, 1970) 403-406); the maximal connected ones are known as Smith graphs. Salem graphs are 'spectrally close' to being cyclotomic, in that nearly all their eigenvalues are in the critical interval [−2, 2]. On the other hand, Salem graphs do not need to be 'combinatorially close' to being cyclotomic: the largest cyclotomic induced subgraph might be comparatively tiny.We define an m-Salem graph to be a connected Salem graph G for which m is minimal such that there exists an induced cyclotomic subgraph of G that has m fewer vertices than G. The 1-Salem subgraphs are both spectrally close and combinatorially close to being cyclotomic. Moreover, every Salem graph contains a 1-Salem graph as an induced subgraph, so these 1-Salem graphs provide some necessary substructure of all Salem graphs. The main result of this paper is a complete combinatorial description of all 1-Salem graphs: in the non-bipartite case there are 25 infinite families and 383 sporadic examples.

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“…Reflexive graphs are also called hyperbolic graphs (see, for example, [7]). Reflexivity of graphs with small cyclomatic number (such as trees [7,10], unicyclic graphs [16], bicyclic graphs [17]) or graphs with other special structure (like cactuses [15]) has already been considered in the literature. Line graphs whose second largest eigenvalue does not exceed 1 are described in [13] (for a more general result, see [14]).…”

Section: Introductionmentioning

confidence: 99%

Reflexive line graphs of trees

et al. 2015

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A graph is reflexive if the second largest eigenvalue of its adjacency matrix is less than or equal to 2. In this paper, we characterize trees whose line graphs are reflexive. It turns out that these trees can be of arbitrary order-they can have either a unique vertex of arbitrary degree or pendant paths of arbitrary lengths, or both. Since the reflexive line graphs are Salem graphs, we also relate some of our results to the Salem (graph) numbers.

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Maximal reflexive cacti with four cycles: The approach via Smith graphs

Cited by 10 publications

References 14 publications

Trees whose second largest eigenvalue does not exceed √ 5+1 / 2

Trees whose second largest eigenvalue does not exceed √ 5+1 / 2

A classification of all 1-Salem graphs

Reflexive line graphs of trees

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