scholarly journals Constructing Fifteen Infinite Classes of Nonregular Bipartite Integral Graphs

10.37236/732 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Ligong Wang ◽  
Cornelis Hoede
Keyword(s):  
Number Theory ◽  
Adjacency Matrix ◽  
Diophantine Equations ◽  
Matrix Theory ◽  
Discrete Math ◽  
Computer Search ◽  
Characteristic Polynomials ◽  
Integral Property ◽  
Open Problems ◽  
Integral Graphs

A graph is called integral if all its eigenvalues (of the adjacency matrix) are integers. In this paper, the graphs $S_1(t)=K_{1,t}$, $S_2(n,t)$, $S_3(m,n,t)$, $S_4(m,n,p,q)$, $S_5(m,n)$, $S_6(m,n,t)$, $S_8(m,n)$, $S_9(m,n,p,q)$, $S_{10}(n)$, $S_{13}(m,n)$, $S_{17}(m, n, p, q)$, $S_{18}(n,p,q,t)$, $S_{19}(m,n,p,t)$, $S_{20}(n,p,q)$ and $S_{21}(m,t)$ are defined. We construct the fifteen classes of larger graphs from the known 15 smaller integral graphs $S_1-S_6$, $S_8-S_{10}$, $S_{13}$, $S_{17}-S_{21}$ (see also Figures 4 and 5, Balińska and Simić, Discrete Math. 236(2001) 13-24). These classes consist of nonregular and bipartite graphs. Their spectra and characteristic polynomials are obtained from matrix theory. We obtain their integral property by using number theory and computer search. All these classes are infinite. They are different from those in the literature. It is proved that the problem of finding such integral graphs is equivalent to solving Diophantine equations. We believe that it is useful for constructing other integral graphs. The discovery of these integral graphs is a new contribution to the search of integral graphs. Finally, we propose several open problems for further study.

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 Eigenvalues and triangles in graphs

2020 ◽  
pp. 1-13
Author(s):  
Huiqiu Lin ◽  
Bo Ning ◽  
Baoyindureng Wu
Keyword(s):  
Bipartite Graph ◽  
Adjacency Matrix ◽  
Matrix Theory ◽  
Stochastic Matrix ◽  
Extremal Graphs ◽  
Open Problems ◽  
Free Graph ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Largest Eigenvalues

Abstract Bollobás and Nikiforov (J. Combin. Theory Ser. B.97 (2007) 859–865) conjectured the following. If G is a Kr+1-free graph on at least r+1 vertices and m edges, then ${\rm{\lambda }}_1^2(G) + {\rm{\lambda }}_2^2(G) \le (r - 1)/r \cdot 2m$ , where λ1 (G)and λ2 (G) are the largest and the second largest eigenvalues of the adjacency matrix A(G), respectively. In this paper we confirm the conjecture in the case r=2, by using tools from doubly stochastic matrix theory, and also characterize all families of extremal graphs. Motivated by classic theorems due to Erdős and Nosal respectively, we prove that every non-bipartite graph G of order n and size m contains a triangle if one of the following is true: (i) ${{\rm{\lambda }}_1}(G) \ge \sqrt {m - 1} $ and $G \ne {C_5} \cup (n - 5){K_1}$ , and (ii) ${{\rm{\lambda }}_1}(G) \ge {{\rm{\lambda }}_1}(S({K_{[(n - 1)/2],[(n - 1)/2]}}))$ and $G \ne S({K_{[(n - 1)/2],[(n - 1)/2]}})$ , where $S({K_{[(n - 1)/2],[(n - 1)/2]}})$ is obtained from ${K_{[(n - 1)/2],[(n - 1)/2]}}$ by subdividing an edge. Both conditions are best possible. We conclude this paper with some open problems.

scholarly journals Normalized Sombor Indices as Complexity Measures of Random Networks

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 976
Author(s):  
R. Aguilar-Sánchez ◽  
J. Méndez-Bermúdez ◽  
José Rodríguez ◽  
José Sigarreta
Keyword(s):  
Random Matrix ◽  
Adjacency Matrix ◽  
Matrix Theory ◽  
Computational Study ◽  
Random Networks ◽  
Geometric Graphs ◽  
Theory Approach ◽  
Complexity Measures ◽  
Random Geometric Graphs ◽  
Highly Correlated

We perform a detailed computational study of the recently introduced Sombor indices on random networks. Specifically, we apply Sombor indices on three models of random networks: Erdös-Rényi networks, random geometric graphs, and bipartite random networks. Within a statistical random matrix theory approach, we show that the average values of Sombor indices, normalized to the order of the network, scale with the average degree. Moreover, we discuss the application of average Sombor indices as complexity measures of random networks and, as a consequence, we show that selected normalized Sombor indices are highly correlated with the Shannon entropy of the eigenvectors of the adjacency matrix.

scholarly journals Upper bounds on the energy of graphs in terms of matching number

2021 ◽  
pp. 16-16
Author(s):  
Saieed Akbari ◽  
Abdullah Alazemi ◽  
Milica Andjelic
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Short Proof ◽  
Upper Bounds ◽  
Vertex Degree ◽  
Maximum Matching ◽  
Matching Number ◽  
Open Problems

The energy of a graph G, ?(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. The matching number ?(G) is the number of edges in a maximum matching. In this paper, for a connected graph G of order n with largest vertex degree ? ? 6 we present two new upper bounds for the energy of a graph: ?(G) ? (n-1)?? and ?(G) ? 2?(G)??. The latter one improves recently obtained bound ?(G) ? {2?(G)?2?e + 1, if ?e is even; ?(G)(? a + 2?a + ?a-2?a), otherwise, where ?e stands for the largest edge degree and a = 2(?e + 1). We also present a short proof of this result and several open problems.

New families of integral graphs

2016 ◽  
Vol 08 (04) ◽  
pp. 1650063
Author(s):  
Indulal Gopalapillai
Keyword(s):  
Adjacency Matrix ◽  
Simple Graph ◽  
Matrix Formula ◽  
Integral Graphs

Let [Formula: see text] be a simple graph with an adjacency matrix [Formula: see text]. Then the eigenvalues of [Formula: see text] are the eigenvalues of [Formula: see text] and form the spectrum, [Formula: see text] of [Formula: see text]. The graph [Formula: see text] is integral if [Formula: see text] consists of only integers. In this paper, we define three new operations on graphs and characterize all integral graphs in the resulting families. The resulting families are denoted by [Formula: see text], and [Formula: see text]. These characterizations allow us to exhibit many new infinite families of integral graphs.

scholarly journals Properties of the characteristic polynomials and spectrum of Pn and Cn

2016 ◽  
Vol 5 (2) ◽  
pp. 132
Author(s):  
Essam El Seidy ◽  
Salah Eldin Hussein ◽  
Atef Mohamed
Keyword(s):  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Simple Graph ◽  
Characteristic Polynomials ◽  
Signless Laplacian Matrix ◽  
Path Graph ◽  
Signless Laplacian ◽  
Vertex Set ◽  
Cycle Graph

We consider a finite undirected and connected simple graph  with vertex set  and edge set .We calculated the general formulas of the spectra of a cycle graph and path graph. In this discussion we are interested in the adjacency matrix, Laplacian matrix, signless Laplacian matrix, normalized Laplacian matrix, and seidel adjacency matrix.

Index, sub-index and sub-factor of groups with interactions to number theory

2019 ◽  
Vol 19 (06) ◽  
pp. 2050101
Author(s):  
M. H. Hooshmand
Keyword(s):  
Number Theory ◽  
Direct Product ◽  
Prime Numbers ◽  
Additive Number Theory ◽  
Open Problems ◽  
Famous Conjecture ◽  
Future Directions ◽  
Left And Right ◽  
The Difference ◽  
Equivalent Conditions

This paper is the first step of a new topic about groups which has close relations and applications to number theory. Considering the factorization of a group into a direct product of two subsets, and since every subgroup is a left and right factor, we observed that the index conception can be generalized for a class of factors. But, thereafter, we found that every subset [Formula: see text] of a group [Formula: see text] has four related sub-indexes: right, left, upper and lower sub-indexes [Formula: see text], [Formula: see text] which agree with the conception index of subgroups, and all of them are equal if [Formula: see text] is a subgroup or normal sub-semigroup of [Formula: see text]. As a result of the topic, we introduce some equivalent conditions to a famous conjecture for prime numbers (“every even number is the difference of two primes”) that one of them is: the prime numbers set is index stable (i.e. all of its sub-indexes are equal) in integers and [Formula: see text]. Index stable groups (i.e. those whose subsets are all index stable) are a challenging subject of the topic with several results and ideas. Regarding the extension of the theory, we give some methods for evaluation of sub-indexes, by using the left and right differences of subsets. At last, we pose many open problems, questions, a proposal for additive number theory, and show some future directions of researches and projects for the theory.

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