Reflexive bipartite regular graphs

Koledin, Tamara; Stanić, Zoran

doi:10.1016/j.laa.2013.07.020

Cited by 9 publications

(7 citation statements)

References 6 publications

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“…Considering its diameter, we conclude thatĠ + is non-bipartite, and thus the tensor product K 2 ×Ġ + must be a connected bipartite and also reflexive graph with 40 vertices. We recall that all reflexive bipartite regular graphs are determined in [4]. According to this reference, there is no such a graph with 40 vertices, and we are done.…”

Section: On Non-complete Srsgs Of Cmentioning

confidence: 99%

“…For r = 6, in the same way, we deduce that K 2 ×Ġ + must be a connected bipartite 5-regular reflexive graph with 24 vertices. According to [4], there are 4 such graphs, all of them with spectrum [5, 2 8 , 1 3 , −1 3 , −2 8 , −5], which means that the least eigenvalue ofĠ + is at least −2. Since ( , λ, µ) = (4, 1, −4), we get m µ = 3, and then using the Courant-Weyl inequalities once again, we get −4 = λ 10 (Ġ) ≥ λ 10 (Ġ + ) + λ 12 (−Ġ − ) = λ 10 (Ġ + ) − 1, which is a contradiction.…”

Section: On Non-complete Srsgs Of Cmentioning

confidence: 99%

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On a class of strongly regular signed graphs

Koledin¹,

Stanić²

2020

Publ. Math. Debrecen

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Let w2(i, j) denote the difference between the numbers of positive and negative walks of length 2 starting at vertex i and terminating at j of a signed graphĠ. Signed graphĠ which is neither homogeneous complete nor totally disconnected is called strongly regular if there exist constants a, b, c such that w2(i, j) = a for i + ∼ j, w2(i, j) = b for i − ∼ j and w2(i, j) = c for i j, respectively. In this paper we consider the class of inhomogeneous strongly regular signed graphs satisfying a = −b, which are either complete or incomplete with c = a+b 2. It occurs that such strongly regular signed graphs have some properties that are analogous to those of strongly regular unsigned graphs. For example, contrary to some other strongly regular signed graphs, they have exactly 3 eigenvalues and their net-degree appears as a simple one.

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Section: On Non-complete Srsgs Of Cmentioning

confidence: 99%

Section: On Non-complete Srsgs Of Cmentioning

confidence: 99%

On a class of strongly regular signed graphs

Koledin¹,

Stanić²

2020

Publ. Math. Debrecen

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“…Here we survey determination of reflexive bipartite regular graphs (for short, RBR graphs) obtained by Koledin and Stanić [10].…”

Section: Appendixmentioning

confidence: 99%

Reflexive cacti: A survey

Mihailovic¹,

Rašajski²,

Stanić

2016

APPL ANAL DISCRETE M

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“…Various classes of multicyclic reflexive graphs have been determined in [10,15,16,17,18,20,21]. Reflexive bipartite regular graphs have been considered in [6].…”

Section: Introductionmentioning

confidence: 99%

Some graph mappings that preserve the sign of λ2 - r

Mihailovic¹,

Rašajski²

2017

APPL ANAL DISCRETE M

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In this article we deal with the sign of λ2 − r, r > 0, where λ2 is the second largest eigenvalue of (adjacency matrix of) a simple graph and present some methods of determining it for some classes of graphs. The main result is a set of graph mappings that preserve the value of sgn (λ2 − r). These mappings induce equivalence relations among involved graphs, thus providing a way to indirectly apply the GRS-theorem (the generalization of so-called RStheorem) to some GRS-undecidable (or RS-undecidable) graphs. To present possible applications, we revisit some of the previous results for reflexive graphs (graphs whose second largest eigenvalue does not exceed 2). We show how maximal reflexive graphs that belong to various families depending on their cyclic structure, can be reduced to RS-decidable graphs in terms of corresponding equivalence relations.

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Reflexive bipartite regular graphs

Cited by 9 publications

References 6 publications

On a class of strongly regular signed graphs

On a class of strongly regular signed graphs

Reflexive cacti: A survey

Some graph mappings that preserve the sign of λ2 - r

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