scholarly journals Laplacian Integral Graphs with Maximum Degree $3$

10.37236/844 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Steve Kirkland
Keyword(s):  
Maximum Degree ◽  
Laplacian Matrix ◽  
Integral Graphs

A graph is said to be Laplacian integral if the spectrum of its Laplacian matrix consists entirely of integers. Using combinatorial and matrix-theoretic techniques, we identify, up to isomorphism, the $21$ connected Laplacian integral graphs of maximum degree $3$ on at least $6$ vertices.

scholarly journals Which non-regular bipartite integral graphs with maximum degree four do not have ±1 as eigenvalues?

2004 ◽  
Vol 286 (1-2) ◽  
pp. 15-24 ◽  
Author(s):  
Krystyna T. Balińska ◽  
Slobodan K. Simić ◽  
Krzysztof T. Zwierzyński
Keyword(s):  
Maximum Degree ◽  
Integral Graphs

scholarly journals Laplacian integral graphs with a given degree sequence constraint

2021 ◽  
Vol 40 (6) ◽  
pp. 1431-1448
Author(s):  
Ansderson Fernandes Novanta ◽  
Carla Silva Oliveira ◽  
Leonardo de Lima
Keyword(s):  
Adjacency Matrix ◽  
Degree Sequence ◽  
Laplacian Matrix ◽  
Bipartite Graphs ◽  
Diagonal Matrix ◽  
Connected Graphs ◽  
Sequence Constraint ◽  
The Matrix ◽  
Vertex Degrees ◽  
Integral Graphs

Let G be a graph on n vertices. The Laplacian matrix of G, denoted by L(G), is defined as L(G) = D(G) −A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the vertex degrees of G. A graph G is said to be L-integral if all eigenvalues of the matrix L(G) are integers. In this paper, we characterize all Lintegral non-bipartite graphs among all connected graphs with at most two vertices of degree larger than or equal to three.

Laplacian integral graphs with a given degreee sequence constraint

2021 ◽  
Author(s):  
Anderson Fernandes Novanta ◽  
Carla Silva Oliveira ◽  
Leonardo Silva de Lima
Keyword(s):  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Bipartite Graphs ◽  
Diagonal Matrix ◽  
Connected Graphs ◽  
Sequence Constraint ◽  
The Matrix ◽  
Vertex Degrees ◽  
Integral Graphs

Let G be a graph on n vertices. The Laplacian matrix of G, denoted by L(G), is defined as L(G) = D(G) − A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the vertex degrees of G. A graph G is said to be L-integral is all eigenvalues of the matrix L(G) are integers. In this paper, we characterize all L-integral non-bipartite graphs among all connected graphs with at most two vertices of degree larger than or equal to three.

scholarly journals The nonregular, bipartite, integral graphs with maximum degree 4. Part I: basic properties

2001 ◽  
Vol 236 (1-3) ◽  
pp. 13-24 ◽  
Author(s):  
Krystyna T. Balińska ◽  
Slobodan K. Simić
Keyword(s):  
Maximum Degree ◽  
Basic Properties ◽  
Integral Graphs

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

2014 ◽  
Vol 8 (1) ◽  
pp. 123-154 ◽  
Author(s):  
Abreu de ◽  
Krystyna Balińska ◽  
Slobodan Simic ◽  
Krzysztof Zwierzyński
Keyword(s):  
Adjacency Matrix ◽  
Maximum Degree ◽  
Complete Solution ◽  
Discrete Math ◽  
Integral Graphs

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. Bali?ska, S.K. Simic, K.T. Zwierzy?ski: Which non-regular 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.

scholarly journals Laplacian integral subcubic signed graphs

2021 ◽  
Vol 37 ◽  
pp. 163-176
Author(s):  
Yaoping Hou ◽  
Dijian Wang
Keyword(s):  
Maximum Degree ◽  
Laplacian Matrix ◽  
Signed Graph ◽  
Signed Graphs

A (signed) graph is called Laplacian integral if all eigenvalues of its Laplacian matrix are integers. In this paper, we determine all connected Laplacian integral signed graphs of maximum degree 3; among these signed graphs,there are two classes of Laplacian integral signed graphs, one contains 4 infinite families of signed graphs and another contains 29 individual signed graphs.

The New Class of Laplacian Integral Graphs

2014 ◽  
Vol 989-994 ◽  
pp. 2643-2646
Author(s):  
Shi Fang Lu ◽  
Jin Yu Zou
Keyword(s):  
Laplacian Matrix ◽  
Characteristic Polynomials ◽  
New Class ◽  
Integral Graphs

Graph is Laplacian integral, if all the eigenvalues of its Laplacian matrix are integral. In this paper, we obtain that the Laplacian characteristic polynomials of graphs by calculation. Characterizes the new class of Laplacian integral graphs .

scholarly journals The p-spectral radius of the Laplacian matrix

2018 ◽  
Vol 12 (2) ◽  
pp. 455-466
Author(s):  
Elizandro Borba ◽  
Eliseu Fritscher ◽  
Carlos Hoppen ◽  
Sebastian Richter
Keyword(s):  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Maximum Degree ◽  
Laplacian Matrix ◽  
Extremal Problems ◽  
Graph Invariants ◽  
Maximum Cut

The p-spectral radius of a graph G=(V,E) with adjacency matrix A is defined as ?(p)(G) = max||x||p=1 xT Ax. This parameter shows connections with graph invariants, and has been used to generalize some extremal problems. In this work, we define the p-spectral radius of the Laplacian matrix L as ?(p)(G) = max||x||p=1 xT Lx. We show that ?(p)(G) relates to invariants such as maximum degree and size of a maximum cut. We also show properties of ?(p)(G) as a function of p, and a upper bound on maxG: |V(G)|=n ?(p)(G) in terms of n = |V| for p > 2, which is attained if n is even.

Non-standard methods for converting numbers from one number system to another

2020 ◽  
Vol 1 (9) ◽  
pp. 28-30
Author(s):  
D. M. Zlatopolski
Keyword(s):  
Binary System ◽  
Maximum Degree ◽  
Number System ◽  
Binary Form ◽  
Binary Number ◽  
Standard Methods ◽  
Optimal Variant ◽  
Decimal System ◽  
Large Numbers ◽  
Independent Work

The article describes a number of little-known methods for translating natural numbers from one number system to another. The first is a method for converting large numbers from the decimal system to the binary system, based on multiple divisions of a given number and all intermediate quotients by 64 (or another number equal to 2n ), followed by writing the last quotient and the resulting remainders in binary form. Then two methods of mutual translation of decimal and binary numbers are described, based on the so-called «Horner scheme». An optimal variant of converting numbers into the binary number system by the method of division by 2 is also given. In conclusion, a fragment of a manuscript from the beginning of the late 16th — early 17th centuries is published with translation into the binary system by the method of highlighting the maximum degree of number 2. Assignments for independent work of students are offered.

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