On the Dα spectral radius of strongly connected digraphs

Let G be a strongly connected digraph with distance matrix D(G) and let Tr(G) be the diagonal matrix with vertex transmissions of G. For any real ? ? [0, 1], define the matrix D?(G) as D?(G) = ?Tr(G) + (1-?)D(G). The D? spectral radius of G is the spectral radius of D?(G). In this paper, we first give some upper and lower bounds for the D? spectral radius of G and characterize the extremal digraphs. Moreover, for digraphs that are not transmission regular, we give a lower bound on the difference between the maximum vertex transmission and the D? spectral radius. Finally, we obtain the D? eigenvalues of the join of certain regular digraphs.

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Restricted connectivity of total digraph

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830916500221 ◽

2016 ◽

Vol 08 (02) ◽

pp. 1650022

Author(s):

Zhenhua Liu ◽

Zhao Zhang

Keyword(s):

Lower Bounds ◽

Upper And Lower Bounds ◽

Vertex Connectivity ◽

Minimum Cardinality ◽

Strongly Connected Digraph ◽

Connected Digraph ◽

Strongly Connected

For a strongly connected digraph [Formula: see text], an arc-cut [Formula: see text] is a [Formula: see text]-restricted arc-cut of [Formula: see text] if [Formula: see text] has at least two strong components of order at least [Formula: see text]. The [Formula: see text]-restricted arc-connectivity [Formula: see text] is the minimum cardinality of all [Formula: see text]-restricted arc-cuts. The [Formula: see text]-restricted vertex-connectivity [Formula: see text] can be defined similarly. In this paper, we provide upper and lower bounds for [Formula: see text] and [Formula: see text] for the total digraph [Formula: see text] of [Formula: see text].

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Difference betwwen the maximum degree and the index of a graph

Proceedings of the National Academy of Sciences of Belarus Physics and Mathematics Series ◽

10.29235/1561-2430-2018-54-4-434-440 ◽

2019 ◽

Vol 54 (4) ◽

pp. 434-440

Author(s):

V. I. Benediktovich

Keyword(s):

Lower Bound ◽

Spectral Radius ◽

Lower Bounds ◽

Maximum Degree ◽

Lower Boundary ◽

Arbitrary Graph ◽

Analogous Statement ◽

Graph Classes ◽

The Difference ◽

A Chain

An algebraic parameter of a graph – a difference between its maximum degree and its spectral radius is considered in this paper. It is well known that this graph parameter is always nonnegative and represents some measure of deviation of a graph from its regularity. In the last two decades, many papers have been devoted to the study of this parameter. In particular, its lower bound depending on the graph order and diameter was obtained in 2007 by mathematician S. M. Cioabă. In 2017 when studying the upper and the lower bounds of this parameter, M. R. Oboudi made a conjecture that the lower bound of a given parameter for an arbitrary graph is the difference between a maximum degree and a spectral radius of a chain. This is very similar to the analogous statement for the spectral radius of an arbitrary graph whose lower boundary is also the spectral radius of a chain. In this paper, the above conjecture is confirmed for some graph classes.

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LANG'S CONJECTURE AND SHARP HEIGHT ESTIMATES FOR THE ELLIPTIC CURVES y2 = x3 + ax

International Journal of Number Theory ◽

10.1142/s1793042113500176 ◽

2013 ◽

Vol 09 (05) ◽

pp. 1141-1170 ◽

Cited By ~ 1

Author(s):

PAUL VOUTIER ◽

MINORU YABUTA

Keyword(s):

Lower Bound ◽

Elliptic Curves ◽

Lower Bounds ◽

Rational Points ◽

Upper And Lower Bounds ◽

Canonical Height ◽

The Difference ◽

Lang's Conjecture ◽

Lang’S Conjecture ◽

Logarithmic Height

For elliptic curves given by the equation Ea : y2 = x3 + ax, we establish the best-possible version of Lang's conjecture on the lower bound for the canonical height of non-torsion rational points along with best-possible upper and lower bounds for the difference between the canonical and logarithmic height.

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On the Generalized Distance Energy of Graphs

Mathematics ◽

10.3390/math8010017 ◽

2019 ◽

Vol 8 (1) ◽

pp. 17 ◽

Cited By ~ 4

Author(s):

Abdollah Alhevaz ◽

Maryam Baghipur ◽

Hilal A. Ganie ◽

Yilun Shang

Keyword(s):

Lower Bounds ◽

Complete Graph ◽

Connected Graph ◽

Distance Matrix ◽

Wiener Index ◽

Diagonal Matrix ◽

Upper And Lower Bounds ◽

Star Graph ◽

Extremal Graphs ◽

Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

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On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

Mathematics ◽

10.3390/math9050512 ◽

2021 ◽

Vol 9 (5) ◽

pp. 512

Author(s):

Maryam Baghipur ◽

Modjtaba Ghorbani ◽

Hilal A. Ganie ◽

Yilun Shang

Keyword(s):

Lower Bounds ◽

Complete Graph ◽

Distance Matrix ◽

Upper And Lower Bounds ◽

Laplacian Eigenvalue ◽

Signless Laplacian ◽

Graph Parameters ◽

Simple Connected Graph ◽

Second Largest Eigenvalue ◽

Reciprocal Distance Matrix

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.

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The normalized distance Laplacian

Special Matrices ◽

10.1515/spma-2020-0114 ◽

2021 ◽

Vol 9 (1) ◽

pp. 1-18

Author(s):

Carolyn Reinhart

Keyword(s):

Spectral Radius ◽

Characteristic Polynomial ◽

Adjacency Matrix ◽

Connected Graph ◽

Distance Matrix ◽

Laplacian Matrix ◽

Diagonal Matrix ◽

The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

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A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

Algorithms ◽

10.3390/a14060164 ◽

2021 ◽

Vol 14 (6) ◽

pp. 164

Author(s):

Tobias Rupp ◽

Stefan Funke

Keyword(s):

Lower Bound ◽

Lower Bounds ◽

Real World ◽

Road Network ◽

Upper And Lower Bounds ◽

Bounded Degree ◽

Shortest Path Routing ◽

Graph Classes ◽

Speed Up ◽

To Come

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

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On the Rank of $p$-Schemes

The Electronic Journal of Combinatorics ◽

10.37236/3097 ◽

2013 ◽

Vol 20 (2) ◽

Author(s):

Fateme Raei Barandagh ◽

Amir Rahnamai Barghi

Keyword(s):

Lower Bound ◽

Prime Number ◽

Lower Bounds ◽

Upper Bound ◽

Upper And Lower Bounds ◽

Minimum Rank

Let $n>1$ be an integer and $p$ be a prime number. Denote by $\mathfrak{C}_{p^n}$ the class of non-thin association $p$-schemes of degree $p^n$. A sharp upper and lower bounds on the rank of schemes in $\mathfrak{C}_{p^n}$ with a certain order of thin radical are obtained. Moreover, all schemes in this class whose rank are equal to the lower bound are characterized and some schemes in this class whose rank are equal to the upper bound are constructed. Finally, it is shown that the scheme with minimum rank in $\mathfrak{C}_{p^n}$ is unique up to isomorphism, and it is a fusion of any association $p$-schemes with degree $p^n$.

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New Bounds for the α-Indices of Graphs

Mathematics ◽

10.3390/math8101668 ◽

2020 ◽

Vol 8 (10) ◽

pp. 1668

Author(s):

Eber Lenes ◽

Exequiel Mallea-Zepeda ◽

Jonnathan Rodríguez

Keyword(s):

Lower Bound ◽

Spectral Radius ◽

Upper Bound ◽

Adjacency Matrix ◽

Independence Number ◽

Minimal Degree ◽

Diagonal Matrix ◽

The Matrix

Let G be a graph, for any real 0≤α≤1, Nikiforov defines the matrix Aα(G) as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and diagonal matrix of degrees of the vertices of G. This paper presents some extremal results about the spectral radius ρα(G) of the matrix Aα(G). In particular, we give a lower bound on the spectral radius ρα(G) in terms of order and independence number. In addition, we obtain an upper bound for the spectral radius ρα(G) in terms of order and minimal degree. Furthermore, for n>l>0 and 1≤p≤⌊n−l2⌋, let Gp≅Kl∨(Kp∪Kn−p−l) be the graph obtained from the graphs Kl and Kp∪Kn−p−l and edges connecting each vertex of Kl with every vertex of Kp∪Kn−p−l. We prove that ρα(Gp+1)<ρα(Gp) for 1≤p≤⌊n−l2⌋−1.

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