Notes on the second largest eigenvalue of a graph

Simić, Slavko; Anđelić, Milica; Fonseca, Carlos M. da; Živković, Dejan

doi:10.1016/j.laa.2014.09.032

Cited by 9 publications

(12 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…First, if α ≥ 5, then E = f (α, 0) + 2q min ≥ 147 >F = 144. From (9) we obtain that α ≥ 3, and next we have: if α = 4 then β ≥ 1; if α = 3 then β ≥ 3. So E = min{f (α, β)} + 2q min ≥ 145 >F = 144.…”

Section: Proposition 44mentioning

confidence: 90%

“…Then α ≥ 1 (by (9)). If α ≥ 2 then E ≥ f (2, 0) + 2q min = 143 >F = 140; if α = 1 then β ≥ 3 (by (9)), and we are done (since f (1, 3) = 103). If min ≥ 6 then E = f (0, 1) + 2q min ≥ 145 >F = 140.…”

Section: Proposition 44mentioning

confidence: 95%

“…If equality holds and if |H| ≥ 2a The next result is taken from [9] (see Theorem 4.3). Since G ∈ S is a reflexive graph we can put some further constraints on it.…”

Section: Further Spectral Toolsmentioning

confidence: 99%

See 2 more Smart Citations

More on non-regular bipartite integral graphs with maximum degree 4 not having ±1 as eigenvalues

Balińska

Simić

et al. 2014

APPL ANAL DISCRETE M

View full text Add to dashboard Cite

A graph is integral if the spectrum (of its adjacency matrix) consists entirely of integers. The problem of determining all non-regular bipartite integral graphs with maximum degree four which do not have ±1 as eigenvalues was posed in K.T. Balińska, S.K. Simić, K.T. Zwierzyński: Which nonregular bipartite integral graphs with maximum degree four do not have ±1 as eigenvalues? Discrete Math., 286 (2004), 15-25. Here we revisit this problem, and provide its complete solution using mostly the theoretical arguments.The paper is dedicated to professor Dobrilo D -. Tošić on occasion of his 81st birthday.

show abstract

Section: Proposition 44mentioning

confidence: 90%

Section: Proposition 44mentioning

confidence: 95%

See 1 more Smart Citation

More on non-regular bipartite integral graphs with maximum degree 4 not having ±1 as eigenvalues

Balińska

Simić

et al. 2014

APPL ANAL DISCRETE M

View full text Add to dashboard Cite

show abstract

“…This fact is clear from the works that can be found in the literature regarding the second-largest eigenvalue of the graph with respect to different graph matrices. For some recent works on the second-largest adjacency eigenvalue, we refer to [14]; for the second-largest Laplacian eigenvalue, we refer to [15]; for the second-largest signless Laplacian eigenvalue, we refer to [16]; for the second-largest distance eigenvalue, we refer to [17]; for the second-largest generalized distance eigenvalue, we refer to [18], and so forth. Motivated by these works, we, in this paper, study the second-largest signless Laplacian reciprocal distance eigenvalue of a connected graph.…”

Section: Introductionmentioning

confidence: 99%

On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

Ghorbani

Ganie²,

Shang

2021

Mathematics

View full text Add to dashboard Cite

The signless Laplacian reciprocal distance matrix for a simple connected graph G is defined as RQ(G)=diag(RH(G))+RD(G). Here, RD(G) is the Harary matrix (also called reciprocal distance matrix) while diag(RH(G)) represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with n vertices, the complete graph Kn and the graph Kn−e obtained from Kn by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue.

show abstract

“…Then, besides the interlacing, one needs more arguments (see, for example, [3, Theorems 0.3 and 0.6, pp. [18][19]). …”

mentioning

confidence: 99%

On the multiplicities of eigenvalues of graphs and their vertex deleted subgraphs: old and new results

Simić

Anđelić

Fonseca

et al. 2015

ELA

View full text Add to dashboard Cite

Simic, Slobodan K.; Andelic, Milica; Da Fonseca, Carlos M.; and Zivkovic, Dejan. (2015) Abstract. Given a simple graph G, let A G be its adjacency matrix. A principal submatrix of A G of order one less than the order of G is the adjacency matrix of its vertex deleted subgraph. It is well-known that the multiplicity of any eigenvalue of A G and such a principal submatrix can differ by at most one. Therefore, a vertex v of G is a downer vertex (neutral vertex, or Parter vertex) with respect to a fixed eigenvalue µ if the multiplicity of µ in A G−v goes down by one (resp., remains the same, or goes up by one). In this paper, we consider the problems of characterizing these three types of vertices under various constraints imposed on graphs being considered, on vertices being chosen and on eigenvalues being observed. By assigning weights to edges of graphs, we generalize our results to weighted graphs, or equivalently to symmetric matrices.

show abstract

Notes on the second largest eigenvalue of a graph

Cited by 9 publications

References 10 publications

More on non-regular bipartite integral graphs with maximum degree 4 not having ±1 as eigenvalues

More on non-regular bipartite integral graphs with maximum degree 4 not having ±1 as eigenvalues

On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

On the multiplicities of eigenvalues of graphs and their vertex deleted subgraphs: old and new results

Contact Info

Product

Resources

About