Empirical sampling of connected graph partitions for redistricting

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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Hamiltonicity of a class of toroidal graphs

Mathematica Slovaca ◽

10.1515/ms-2017-0367 ◽

2020 ◽

Vol 70 (2) ◽

pp. 497-503

Author(s):

Dipendu Maity ◽

Ashish Kumar Upadhyay

Keyword(s):

Connected Graph ◽

Hamiltonian Cycle ◽

Partial Solution ◽

The Other ◽

Hamiltonian Cycles ◽

The Face ◽

Toroidal Graphs

Abstract If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. There are eleven types of semi-equivelar maps on the torus. In 1972 Altshuler has presented a study of Hamiltonian cycles in semi-equivelar maps of three types {36}, {44} and {63} on the torus. In this article we study Hamiltonicity of semi-equivelar maps of the other eight types {33, 42}, {32, 41, 31, 41}, {31, 61, 31, 61}, {34, 61}, {41, 82}, {31, 122}, {41, 61, 121} and {31, 41, 61, 41} on the torus. This gives a partial solution to the well known Conjecture that every 4-connected graph on the torus has a Hamiltonian cycle.

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Privacy Preserving Distributed Summation in a Connected Graph

IFAC-PapersOnLine ◽

10.1016/j.ifacol.2020.12.1677 ◽

2020 ◽

Vol 53 (2) ◽

pp. 3445-3450

Author(s):

Katrine Tjell ◽

Rafael Wisniewski

Keyword(s):

Connected Graph ◽

Privacy Preserving

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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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On minimum algebraic connectivity of graphs whose complements are bicyclic

Open Mathematics ◽

10.1515/math-2019-0119 ◽

2019 ◽

Vol 17 (1) ◽

pp. 1490-1502 ◽

Cited By ~ 1

Author(s):

Jia-Bao Liu ◽

Muhammad Javaid ◽

Mohsin Raza ◽

Naeem Saleem

Keyword(s):

Connected Graph ◽

Diffusion Processes ◽

Laplacian Matrix ◽

Group Differences ◽

Algebraic Connectivity ◽

Connected Graphs ◽

Bicyclic Graph ◽

Smallest Eigenvalue ◽

Unique Graph ◽

The Stability

Abstract The second smallest eigenvalue of the Laplacian matrix of a graph (network) is called its algebraic connectivity which is used to diagnose Alzheimer’s disease, distinguish the group differences, measure the robustness, construct multiplex model, synchronize the stability, analyze the diffusion processes and find the connectivity of the graphs (networks). A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. In this paper, firstly the unique graph with a minimum algebraic connectivity is characterized in the class of connected graphs whose complements are bicyclic with exactly three cycles. Then, we find the unique graph of minimum algebraic connectivity in the class of connected graphs $\begin{array}{} {\it\Omega}^c_{n}={\it\Omega}^c_{1,n}\cup{\it\Omega}^c_{2,n}, \end{array}$ where $\begin{array}{} {\it\Omega}^c_{1,n} \end{array}$ and $\begin{array}{} {\it\Omega}^c_{2,n} \end{array}$ are classes of the connected graphs in which the complement of each graph of order n is a bicyclic graph with exactly two and three cycles, respectively.

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The Maximum Genus on a 3-Vertex-Connected Graph

Graphs and Combinatorics ◽

10.1007/s003730070022 ◽

2000 ◽

Vol 16 (2) ◽

pp. 159-164 ◽

Cited By ~ 3

Author(s):

Yuangqiu Huang

Keyword(s):

Connected Graph ◽

Maximum Genus

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THE GENERAL POSITION NUMBER OF THE CARTESIAN PRODUCT OF TWO TREES

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972720001276 ◽

2020 ◽

pp. 1-10

Author(s):

JING TIAN ◽

KEXIANG XU ◽

SANDI KLAVŽAR

Keyword(s):

Shortest Path ◽

Connected Graph ◽

General Position ◽

Cartesian Product

Abstract The general position number of a connected graph is the cardinality of a largest set of vertices such that no three pairwise-distinct vertices from the set lie on a common shortest path. In this paper it is proved that the general position number is additive on the Cartesian product of two trees.

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Data-driven graph drawing techniques with applications for conveyor systems

Journal of Mathematics in Industry ◽

10.1186/s13362-020-00092-2 ◽

2020 ◽

Vol 10 (1) ◽

Author(s):

Simone Göttlich ◽

Sven Spieckermann ◽

Stephan Stauber ◽

Andrea Storck

Keyword(s):

Real World ◽

Connected Graph ◽

Stress Function ◽

Graph Drawing ◽

Point Of View ◽

Data Driven ◽

Challenging Problem ◽

Conveyor System ◽

System Graph ◽

Real World Problems

AbstractThe visualization of conveyor systems in the sense of a connected graph is a challenging problem. Starting from communication data provided by the IT system, graph drawing techniques are applied to generate an appealing layout of the conveyor system. From a mathematical point of view, the key idea is to use the concept of stress majorization to minimize a stress function over the positions of the nodes in the graph. Different to the already existing literature, we have to take care of special features inspired by the real-world problems.

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Hamiltonian Cycles in Strong Products of Graphs

Canadian Mathematical Bulletin ◽

10.4153/cmb-1979-037-0 ◽

1979 ◽

Vol 22 (3) ◽

pp. 305-309 ◽

Cited By ~ 5

Author(s):

J. C. Bermond ◽

A. Germa ◽

M. C. Heydemann

Keyword(s):

Connected Graph ◽

Hamiltonian Cycles ◽

Strong Product

Abstract. Let denote the graph (k times) where is the strong product of the two graphs G and H. In this paper we prove the conjecture of J. Zaks [3]: For every connected graph G with at least two vertices there exists an integer k = k(G) for which the graph is hamiltonian.

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