Adjacency Spectrum Research Articles

Adjacency Spectrum of Reduced Power Graphs of Finite Groups

Adjacency Spectrum of Reduced Power Graphs of Finite Groups

A generalization of Fiedler's lemma and its applications

In this article, taking a Fiedler's result on the spectrum of a matrix formed from two symmetric matrices as a motivation, we deduce a more general result on the eigenvalues of a matrix, which form from n symmetric matrices. As an important application, we obtain the adjacency spectra, Laplacian spectra and signless Laplacian spectra of a graph with a particular almost equitable partition.

The adjacency spectra of some families of minimally connected prime graphs

The adjacency spectra of some families of minimally connected prime graphs

Degree based energy and spectral radius of a graph with self-loops

Let GX be a graph obtained from a simple graph G by attaching a self-loop at each vertex of X ⊆ V ( G ) . The general extended adjacency matrix for the graph GX is defined and the bounds for the degree based energy of the graph GX are obtained. The study extends the notion of degree based energy of simple graphs to graphs with self-loops. For the graph GX of order n and size m with σ self-loops, the adjacency energy, E ( G X ) ≥ 4 mn + 2 σ ( n − σ ) n . The spectral radius ρ ( G X ) of its adjacency matrix is always less than or equal to 1 2 [ − 1 + 1 + 4 ( 2 m + σ + Δ − 1 ) ] , where Δ is the maximum degree in the graph GX and the equality conditions are given for 0 < σ < n . Few more bounds for ρ ( G X ) are also obtained. The study shows that, the spectral radius f max ( G X ) of its extended adjacency matrix satisfies: min i ∼ j F ij ( 2 m + σ ) n ≤ f max ( G X ) ≤ max i ∼ j F ij 2 ( 2 m + σ ) . We conclude the article by computing the extended adjacency spectrum of complete graph and complete bipartite graphs with self-loops.

Universal adjacency spectrum of the cozero-divisor graph and its complement on a finite commutative ring with unity

For a finite simple undirected graph [Formula: see text], the universal adjacency matrix [Formula: see text] is a linear combination of the adjacency matrix [Formula: see text], the degree diagonal matrix [Formula: see text], the identity matrix [Formula: see text] and the all-ones matrix [Formula: see text], that is [Formula: see text], where [Formula: see text] and [Formula: see text]. The cozero-divisor graph [Formula: see text] of a finite commutative ring [Formula: see text] with unity is a simple undirected graph with the set of all nonzero nonunits of [Formula: see text] as vertices and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. In this paper, we study structural properties of [Formula: see text] by defining an equivalence relation on its vertex set in terms of principal ideals of the ring [Formula: see text]. Then we obtain the universal adjacency eigenpairs of [Formula: see text] and its complement, and as a consequence one may obtain several spectra like the adjacency, Seidel, Laplacian, signless Laplacian, normalized Laplacian, generalized adjacency and convex linear combination of the adjacency and degree diagonal matrix of [Formula: see text] and [Formula: see text] in an unified way. Moreover, we get the universal adjacency eigenpairs of the cozero-divisor graph and its complement for a reduced ring and the ring of integers modulo [Formula: see text] in a simpler form.

The Subset-Strong Product of Graphs

Abstract In this paper, we introduce the subset-strong product of graphs and give a method for calculating the adjacency spectrum of this product. In addition, exact expressions for the first and second Zagreb indices of the subset-strong products of two graphs are reported. Examples are provided to illustrate the applications of this product in some growing graphs and complex networks.

On the spectra and spectral radii of token graphs

Let G be a graph on n vertices. The k-token graph (or symmetric k-th power) of G, denoted by Fk(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(G)$$\\end{document}, has as vertices the nk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${n\\atopwithdelims ()k}$$\\end{document}k-subsets of vertices from G, and two vertices are adjacent when their symmetric difference is a pair of adjacent vertices in G. In particular, Fk(Kn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(K_n)$$\\end{document} is the Johnson graph J(n, k), which is a distance-regular graph used in coding theory. In this paper, we present some results concerning the (adjacency and Laplacian) spectrum of Fk(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(G)$$\\end{document} in terms of the spectrum of G. For instance, when G is walk-regular, an exact value for the spectral radius ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho $$\\end{document} (or maximum eigenvalue) of Fk(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(G)$$\\end{document} is obtained. When G is distance-regular, other eigenvalues of its 2-token graph are derived using the theory of equitable partitions. A generalization of Aldous’ spectral gap conjecture (which is now a theorem) is proposed.

Bipartite graphs with all but two eigenvalues equal to 0 and ±1

In this paper we study the class G of all connected bipartite graphs whose adjacency spectrum, apart from the maximum and the minimum eigenvalue, just contains 0 and ±1. It turns out that G consists of five infinite families, each of them containing an infinite subfamily of integral graphs, and seven sporadic graphs. Moreover, we find all graphs in G determined by their spectrum and identify the cospectral mates of the remaining ones.

NEPS of complex unit gain graphs

A complex unit gain graph (or $\mathbb T$-gain graph) is a gain graph with gains in $\mathbb T$, the multiplicative group of complex units. Extending a classical construction for simple graphs due to Cvektovic, suitably defined noncomplete extended $p$-sums (NEPS, for short) of $\mathbb T$-gain graphs are considered in this paper. Structural properties of NEPS like balance and some spectral properties and invariants of their adjacency and Laplacian matrices are investigated, including the energy and the possible symmetry of the adjacency spectrum. It is also shown how NEPS are useful to obtain infinitely many integral graphs from the few at hands.Moreover, it is studied how NEPS of $\mathbb T$-gain graphs behave with respect to the property of being nut, i.e., having $0$ as simple adjacency eigenvalue and nowhere zero $0$-eigenvectors. Finally, a family of new products generalizing NEPS is introduced, and their few first spectral properties explored.

Universal adjacency spectrum of the commuting and non-commuting graphs on AC-groups

For a simple undirected graph [Formula: see text] and [Formula: see text], [Formula: see text], the universal adjacency matrix [Formula: see text], where [Formula: see text] is the identity matrix, [Formula: see text] is the all-ones matrix, and [Formula: see text] and [Formula: see text] are the adjacency and degree diagonal matrices of [Formula: see text], respectively. For a finite non-abelian group [Formula: see text] and [Formula: see text], the commuting graph [Formula: see text] is a simple undirected graph with vertex set [Formula: see text] and the adjacency rule is commuting. The non-commuting graph of [Formula: see text] is the complement of [Formula: see text]. An AC-group is a non-abelian group such that the centralizer of every non-central element is abelian. In this paper, we obtain the universal adjacency eigenpairs of the commuting and non-commuting graphs for an arbitrary AC-group [Formula: see text] with [Formula: see text] or [Formula: see text]. Moreover, we determine the eigenpairs of [Formula: see text] when [Formula: see text] is a specific kind of group.

On the divisibility of H-shape trees and their spectral determination

A graph G is divisible by a graph H if the characteristic polynomial of G is divisible by that of H. In this paper, a necessary and sufficient condition for recursive graphs to be divisible by a path is used to show that the H-shape graph P2,2;n−42,n−7, known to be (for n large enough) the minimizer of the spectral radius among the graphs of order n and diameter n−5, is determined by its adjacency spectrum if and only if n≠10,13,15.

On the spectrum, energy and Laplacian energy of graphs with self-loops

The total energy of a conjugated molecule's π-electrons is a quantum-theoretical feature that has been known since the 1930s. It is determined using the Huckel tight-binding molecular orbital (HMO) method. In 1978, a modified definition of the total π-electron energy was introduced, which is now known as the graph energy. It is calculated by summing the absolute values of the eigenvalues of the adjacency matrix. Quiet Recently in the year 2022, Gutman extended the concept of conjugated systems to hetero-conjugated systems which is the extension of ordinary graph energy to energy of graph with self loops. Let Gσ be an order (vertices) ‘p’ graph with ‘q’ edges and σ− self loops. The adjacency matrix of Gσ is defined by A(Gσ)=(aij) if vi∼adjvj then aij=1; if vi=vj where vi∈Vσ then aii=1 and zero otherwise, where Vσ represents the set of all vertices with loops. Then the energy of graph with self loops is defined as E(Gσ)=∑|λi−σ/p|. In this paper, we aim to analyze the adjacency and Laplacian spectra of certain non-simple standard graphs that contain self-loops. We also calculate the energy and Laplacian energy of these graphs with loops. Furthermore, we derive lower bounds for the energy of any graph containing loops and develop a MATLAB algorithm to calculate these quantities for selected non-simple standard graphs with self-loops. Our study evaluates the strength of a graph by considering the presence of loops, which are edges that connect a vertex to itself. This approach accounts for the impact of each vertex on the entire structure of the graph. By analyzing the energy of a graph with loops, we can gain a better understanding of its distinctive characteristics and behavior.

Distance and adjacency spectra and eigenspaces for three (di)graph lifts: A unified approach

By the induced character theory, we first establish the irreducible decomposition of a permutation representation. Then, using it as a unified tool, we derive a decomposition formula for the distance spectrum and eigenspace of a regular lift of a graph, the adjacency spectra and eigenspaces of a relative lift and a ramified uniform lift of a digraph. The latter results are a complement of Dalfó et al. (2021) [6] and Deng et al. (2007) [8].

Adjacency spectra of semigraphs

In this paper, we define the adjacency matrix of a semigraph. We give the conditions for a matrix to be semigraphical and give an algorithm to construct a semigraph from the semigraphical matrices. We derive lower and upper bounds for largest eigenvalues. We study the eigenvalues of adjacency matrix of two types of star semigraphs.

On adjacency and Laplacian cospectral non-isomorphic signed graphs

Let Γ = (G,σ) be a signed graph, where σ is the sign function on the edges of G. In this paper, we use the operation of partial transpose to obtain non-isomorphic Laplacian cospectral signed graphs. We will introduce two new operations on signed graphs. These operations will establish a relationship between the adjacency spectrum of one signed graph with the Laplacian spectrum of another signed graph. As an application, these new operations will be utilized to construct several pairs of cospectral non-isomorphic signed graphs. Finally, we construct integral signed graphs.

SPECTRA OF GRAPH OPERATIONS BASED ON SPLITTING GRAPH

The splitting graph $ SP(G) $ of a graph $ G $ is the graph obtained from $ G $ by taking a new vertex $ u' $ for each $ u \in V(G) $ and joining $ u' $ to all vertices of $ G $ adjacent to $ u $. For a connected regular graph $ G_1 $ and an arbitrary regular graph $ G_2 $, we determine the adjacency (respectively, Laplacian and signless Laplacian) spectra of two types of graph operations on $ G_1 $ and $ G_2 $ involving the $ SP $-graph of $ G_1 $. Moreover, applying these results we construct some non-regular simultaneous cospectral graphs for the adjacency, Laplacian and signless Laplacian matrices, and compute the Kirchhoff index and the number of spanning trees of the newly constructed graphs.

Universal adjacency spectrum of the looped zero divisor graph for a finite commutative ring with unity

For a finite undirected looped graph [Formula: see text], the universal adjacency matrix [Formula: see text] is a linear combination of the adjacency matrix [Formula: see text], the degree matrix [Formula: see text], the identity matrix [Formula: see text] and the all-ones matrix [Formula: see text], that is [Formula: see text], where [Formula: see text] and [Formula: see text]. For a finite commutative ring [Formula: see text] with unity, the looped zero divisor graph [Formula: see text] is an undirected graph with the set of all nonzero zero divisors of [Formula: see text] as vertices and two vertices (not necessarily distinct) [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we study some structural properties of [Formula: see text] by defining an equivalence relation on its vertex set. Then we obtain the universal adjacency eigenpairs of [Formula: see text], and as a consequence several spectra like the adjacency, Seidel, Laplacian, signless Laplacian, normalized Laplacian, generalized adjacency and convex linear combination of the adjacency and degree matrix of [Formula: see text] can be obtained in a unified way. Moreover, we get the structural properties and the universal adjacency eigenpairs of the looped zero divisor graph of a reduced ring in a simpler form.

The spectra of digraphs with Morita equivalent C⁎-algebras

Eilers et al. have recently completed the geometric classification of unital digraph C⁎-algebras up to Morita equivalence using a set of moves on the corresponding digraphs. We explore the question of whether these moves preserve the nonzero elements of the spectrum of a finite digraph, which in this paper is allowed to have loops and parallel edges. We consider several different digraph spectra that have been studied in the literature, answering this question for the Laplace and adjacency spectra, their skew counterparts, the symmetric adjacency spectrum, the adjacency spectrum of the line digraph, the Hermitian adjacency spectrum, and the normalized Laplacian, considering in most cases two ways that these spectra can be defined in the presence of parallel edges. We show that the adjacency spectra of the digraph and line digraph are preserved by a subset of the moves, and the skew adjacency and Laplace spectra are preserved by the Cuntz splice. We give counterexamples to show that the other spectra are not preserved by the remaining moves. The same results hold if one restricts to the class of strongly connected digraphs.

Universal Adjacency Spectrum of the Looped Zero Divisor Graph of $$\mathbb {Z}_n$$

Universal Adjacency Spectrum of the Looped Zero Divisor Graph of $$\mathbb {Z}_n$$

On normalized distance Laplacian eigenvalues of graphs and applications to graphs defined on groups and rings

The normalized distance Laplacian matrix of a connected graph $ G $, denoted by $ D^{\mathcal{L}}(G) $, is defined by $ D^{\mathcal{L}}(G)=Tr(G)^{-1/2}D^L(G)Tr(G)^{-1/2}, $ where $ D(G) $ is the distance matrix, the $D^{L}(G)$ is the distance Laplacian matrix and $ Tr(G)$ is the diagonal matrix of vertex transmissions of $ G. $ The set of all eigenvalues of $ D^{\mathcal{L}}(G) $ including their multiplicities is the normalized distance Laplacian spectrum or $ D^{\mathcal{L}} $-spectrum of $G$. In this paper, we find the $ D^{\mathcal{L}} $-spectrum of the joined union of regular graphs in terms of the adjacency spectrum and the spectrum of an auxiliary matrix. As applications, we determine the $ D^{\mathcal{L}} $-spectrum of the graphs associated with algebraic structures. In particular, we find the $ D^{\mathcal{L}} $-spectrum of the power graphs of groups, the $ D^{\mathcal{L}} $-spectrum of the commuting graphs of non-abelian groups and the $ D^{\mathcal{L}} $-spectrum of the zero-divisor graphs of commutative rings. Several open problems are given for further work.