Minimal polynomial of Cayley graph adjacency matrix for Boolean functions

2008 ◽  
Vol 11 (2) ◽  
pp. 201-207
Author(s):  
Michel Mitton
Keyword(s):  
Cayley Graph ◽  
Adjacency Matrix ◽  
Boolean Functions ◽  
Minimal Polynomial

scholarly journals On the Characteristic Polynomial of $n$-Cayley Digraphs

10.37236/3105 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Majid Arezoomand ◽  
Bijan Taeri
Keyword(s):  
Cayley Graph ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Cayley Graphs ◽  
Irreducible Representations ◽  
Cayley Digraph ◽  
Semiregular Subgroup

A digraph $\Gamma$ is called $n$-Cayley digraph over a group $G$‎, ‎if there exists a semiregular subgroup $R_G$ of Aut$(\Gamma)$ isomorphic to $G$ with $n$ orbits‎. ‎In this paper‎, ‎we represent the adjacency matrix of $\Gamma$ as a diagonal block‎ ‎matrix in terms of irreducible representations of $G$ and determine its characteristic polynomial‎. ‎As corollaries of this result we find‎:  ‎the spectrum of  semi-Cayley graphs over abelian groups‎, ‎a relation between the characteristic polynomial of an $n$-Cayley graph and its complement‎, ‎and   the spectrum of‎ ‎Calye graphs over groups with cyclic subgroups‎. ‎Finally we determine the eigenspace of $n$-Cayley digraphs and their main eigenvalues‎.

scholarly journals Some Algebraic Properties of a Class of Integral Graphs Determined by Their Spectrum

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Jia-Bao Liu ◽  
S. Morteza Mirafzal ◽  
Ali Zafari
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Adjacency Matrix ◽  
Prime Integer ◽  
Integral Graph ◽  
Algebraic Properties ◽  
Integral Graphs

Let Γ = V , E be a graph. If all the eigenvalues of the adjacency matrix of the graph Γ are integers, then we say that Γ is an integral graph. A graph Γ is determined by its spectrum if every graph cospectral to it is in fact isomorphic to it. In this paper, we investigate some algebraic properties of the Cayley graph Γ = Cay ℤ n , S , where n = p m ( p is a prime integer and m ∈ ℕ ) and S = a ∈ ℤ n | a , n = 1 . First, we show that Γ is an integral graph. Also, we determine the automorphism group of Γ . Moreover, we show that Γ and K v ▽ Γ are determined by their spectrum.

scholarly journals A Classification of Ramanujan Unitary Cayley Graphs

10.37236/478 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Andrew Droll
Keyword(s):  
Cayley Graph ◽  
Regular Graph ◽  
Adjacency Matrix ◽  
Complete Characterization ◽  
Absolute Value ◽  
Ramanujan Graph ◽  
Vertex Set ◽  
Unitary Cayley Graph

The unitary Cayley graph on $n$ vertices, $X_n$, has vertex set ${\Bbb Z}/{n\Bbb Z}$, and two vertices $a$ and $b$ are connected by an edge if and only if they differ by a multiplicative unit modulo $n$, i.e. ${\rm gcd}(a-b,n) = 1$. A $k$-regular graph $X$ is Ramanujan if and only if $\lambda(X) \leq 2\sqrt{k-1}$ where $\lambda(X)$ is the second largest absolute value of the eigenvalues of the adjacency matrix of $X$. We obtain a complete characterization of the cases in which the unitary Cayley graph $X_n$ is a Ramanujan graph.

scholarly journals ON THE CAYLEY GRAPHS OF BOOLEAN FUNCTIONS

2020 ◽  
pp. 873
Author(s):  
Lotfallah Pourfaray ◽  
Modjtaba Ghorbani
Keyword(s):  
Automorphism Group ◽  
Boolean Function ◽  
Vector Space ◽  
Cayley Graph ◽  
Boolean Functions ◽  
Cayley Graphs ◽  
Complete Characterization ◽  
Walsh Spectrum

A Boolean function is a function $f:\Bbb{Z}_n^2 \rightarrow \{0,1\}$ and we denote the set of all $n$-variable Boolean functions by $BF_n$. For $f\in BF_n$ the vector $[{\rm W}_f(a_0),\ldots,{\rm W}_f(a_{2n-1})]$ is called the Walsh spectrum of $f$, where ${\rm W}_f(a)= \sum_{x\in V} (-1)^{f(x) \oplus ax}$, where $V_n$ is the vector space of dimension $n$ over the two-element field $F_2$. In this paper, we shall consider the Cayley graph $\Gamma_f$ associated with a Boolean function $f$. We shall also find a complete characterization of the bent Boolean functions of order $16$ and determine the spectrum of related Cayley graphs.In addition, we shall enumerate all orbits of the action of automorphism group on the set $BF_n$. 

scholarly journals The rank of Pseudo walk matrices: controllable and recalcitrant pairs

2020 ◽  
Vol 3 (3) ◽  
pp. 41-52
Author(s):  
Alexander Farrugia ◽  
Keyword(s):  
Lower Bound ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Minimal Polynomial ◽  
Full Rank ◽  
Gram Matrix ◽  
Largest Eigenvalue ◽  
The Largest Eigenvalue

A pseudo walk matrix \(\mathbf{W}_\mathbf{v}\) of a graph \(G\) having adjacency matrix \(\mathbf{A}\) is an \(n\times n\) matrix with columns \(\mathbf{v},\mathbf{A}\mathbf{v},\mathbf{A}^2\mathbf{v},\ldots,\mathbf{A}^{n-1}\mathbf{v}\) whose Gram matrix has constant skew diagonals, each containing walk enumerations in \(G\). We consider the factorization over \(\mathbb{Q}\) of the minimal polynomial \(m(G,x)\) of \(\mathbf{A}\). We prove that the rank of \(\mathbf{W}_\mathbf{v}\), for any walk vector \(\mathbf{v}\), is equal to the sum of the degrees of some, or all, of the polynomial factors of \(m(G,x)\). For some adjacency matrix \(\mathbf{A}\) and a walk vector \(\mathbf{v}\), the pair \((\mathbf{A},\mathbf{v})\) is controllable if \(\mathbf{W}_\mathbf{v}\) has full rank. We show that for graphs having an irreducible characteristic polynomial over \(\mathbb{Q}\), the pair \((\mathbf{A},\mathbf{v})\) is controllable for any walk vector \(\mathbf{v}\). We obtain the number of such graphs on up to ten vertices, revealing that they appear to be commonplace. It is also shown that, for all walk vectors \(\mathbf{v}\), the degree of the minimal polynomial of the largest eigenvalue of \(\mathbf{A}\) is a lower bound for the rank of \(\mathbf{W}_\mathbf{v}\). If the rank of \(\mathbf{W}_\mathbf{v}\) attains this lower bound, then \((\mathbf{A},\mathbf{v})\) is called a recalcitrant pair. We reveal results on recalcitrant pairs and present a graph having the property that \((\mathbf{A},\mathbf{v})\) is neither controllable nor recalcitrant for any walk vector \(\mathbf{v}\).

scholarly journals The Energy of Cayley Graphs for Symmetric Groups of Order 24

2020 ◽  
pp. 1-6
Author(s):  
Amira Fadina Ahmad Fadzil ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Adjacency Matrix ◽  
Cayley Graphs ◽  
Symmetric Groups ◽  
Finite Graph ◽  
Energy Of Graph ◽  
A Finite Group

A Cayley graph of a finite group G with respect to a subset S of G is a graph where the vertices of the graph are the elements of the group and two distinct vertices x and y are adjacent to each other if xy−1 is in the subset S. The subset of the Cayley graph is inverse closed and does not include the identity of the group. For a simple finite graph, the energy of a graph can be determined by summing up the positive values of the eigenvalues of the adjacency matrix of the graph. In this paper, the graph being studied is the Cayley graph of symmetric group of order 24 where S is the subset of S4 of valency up to two. From the Cayley graphs, the eigenvalues are calculated by constructing the adjacency matrix of the graphs and by using some properties of special graphs. Finally, the energy of the respected Cayley graphs is computed and presented. Keywords: energy of graph; cayley graph; symmetric groups

scholarly journals The Energy of Cayley Graphs for a Generating Subset of the Dihedral Groups

MATEMATIKA ◽  
2019 ◽  
Vol 35 (3) ◽  
Author(s):  
Amira Fadina Ahmad Fadzil ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Finite Group ◽  
Cayley Graph ◽  
Adjacency Matrix ◽  
Dihedral Group ◽  
Cayley Graphs ◽  
Dihedral Groups ◽  
Specific Subset ◽  
A Finite Group

Let G be a finite group and S be a subset of G where S does not include the identity of G and is inverse closed. A Cayley graph of a group G with respect to the subset S is a graph where its vertices are the elements of G and two vertices a and b are connected if ab^(−1) is in the subset S. The energy of a Cayley graph is the sum of all absolute values of the eigenvalues of its adjacency matrix. In this paper, we consider a specific subset S = {b, ab, . . . , a^(n−1)b} for dihedral group of order 2n, where n is greater or equal to 3 and find the Cayley graph with respect to the set. We also calculate the eigenvalues and compute the energy of the respected Cayley graphs. Finally, the generalization of the energy of the respected Cayley graphs is found.

scholarly journals Improved Expansion of Random Cayley Graphs

2004 ◽  
Vol Vol. 6 no. 2 ◽  
Author(s):  
Po-Shen Loh ◽  
Leonard J. Schulman
Keyword(s):  
Cayley Graph ◽  
Adjacency Matrix ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Expected Value ◽  
Small Eigenvalue ◽  
Irreducible Representations ◽  
Absolute Value ◽  
Graph Expansion ◽  
International Audience

International audience In Random Cayley Graphs and Expanders, N. Alon and Y. Roichman proved that for every ε > 0 there is a finite c(ε ) such that for any sufficiently large group G, the expected value of the second largest (in absolute value) eigenvalue of the normalized adjacency matrix of the Cayley graph with respect to c(ε ) log |G| random elements is less than ε . We reduce the number of elements to c(ε )log D(G) (for the same c), where D(G) is the sum of the dimensions of the irreducible representations of G. In sufficiently non-abelian families of groups (as measured by these dimensions), log D(G) is asymptotically (1/2)log|G|. As is well known, a small eigenvalue implies large graph expansion (and conversely); see Tanner84 and AlonMilman84-2,AlonMilman84-1. For any specified eigenvalue or expansion, therefore, random Cayley graphs (of sufficiently non-abelian groups) require only half as many edges as was previously known.

scholarly journals Integral Mixed Cayley Graphs over Abelian Groups

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Monu Kadyan ◽  
Bikash Bhattacharjya
Keyword(s):  
Abelian Group ◽  
Cayley Graph ◽  
Adjacency Matrix ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Mixed Graph ◽  
Vertex Set ◽  
Gamma S

A mixed graph is said to be integral if all the eigenvalues of its Hermitian adjacency matrix are integer. Let $\Gamma$ be an abelian group. The mixed Cayley graph $Cay(\Gamma,S)$ is a mixed graph on the vertex set $\Gamma$ and edge set $\left\{ (a,b): b-a\in S \right\}$, where $0\not\in S$. We characterize integral mixed Cayley graph $Cay(\Gamma,S)$ over an abelian group $\Gamma$ in terms of its connection set $S$.

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