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The spectral decomposition of graph adjacency matrices is an essential ingredient in the design of graph signal processing (GSP) techniques. When the adjacency matrix has multi-dimensional eigenspaces, it is desirable to base GSP constructions on a particular eigenbasis that better reflects the graph’s symmetries. In this paper, we provide an explicit and detailed representation-theoretic account for the spectral decomposition of the adjacency matrix of a weighted Cayley graph. Our method applies to all weighted Cayley graphs, regardless of whether they are quasi-Abelian, and offers detailed descriptions of eigenvalues and eigenvectors derived from the coefficient functions of the representations of the underlying group. Next, we turn our attention to constructing frames on Cayley graphs. Frames are overcomplete spanning sets that ensure stable and potentially redundant systems for signal reconstruction. We use our proposed eigenbases to build frames that are suitable for developing signal processing on Cayley graphs. These are the Frobenius–Schur frames and Cayley frames, for which we provide a characterization and a practical recipe for their construction. [ABSTRACT FROM AUTHOR]

In this work, we provide two applications of the eigenvalues of the unitary Cayley graphs over matrix rings over finite fields. For a ring R, $ J_R $ J R denotes the Jacobson radical of R. In 2012, Kiani and Aghaei conjectured that if R and S are finite rings and their unitary Cayley graphs are isomorphic, then $ R/J_R $ R / J R and $ S/J_S $ S / J S are isomorphic. If this conjecture holds and $ J_R = \{0\} $ J R = { 0 } , then R is characterized by its unitary Cayley graph, and we say that R is a ring determined by unitary Cayley graphs. Kiani and Aghaei showed that every finite commutative ring and $ \operatorname {\textnormal {M}}_n(\operatorname {\mathbb {F}}_q) $ M n ⁡ ( F q) are such rings. We examine the eigenvalues of $ \operatorname {\textnormal {M}}_n(\operatorname {\mathbb {F}}_q) $ M n ⁡ ( F q) and obtain many new families of rings determined by unitary Cayley graphs. In 2021, Podestá and Videla characterized all finite commutative rings R such that the graphs in the triple $ \{ \textnormal {C}_{R}, \textnormal {C}^+_R, \bar {\textnormal {C}}_R\} $ { C R , C R + , C ¯ R } are mutually equienergetic non-isospectral and Ramanujan where $ \textnormal {C}^+_R $ C R + is the unitary Cayley sum graph. We use the eigenvalues of the graph $ \textnormal {C}_{\operatorname {\textnormal {M}}_n(\operatorname {\mathbb {F}}_q)} $ C M n ⁡ ( F q) and some observation on the number of matrices of the given rank to extend Podestá and Videla's results by working on the finite non-commutative ring $ \operatorname {\mathcal {R}} $ R being a product of matrix rings over local rings. [ABSTRACT FROM AUTHOR]

In this article, we find the Randić spectrum of the weakly zero-divisor graph of a finite commutative ring [Formula: see text] with identity [Formula: see text], denoted as [Formula: see text], where [Formula: see text] is taken as the ring of integers modulo [Formula: see text]. The weakly zero-divisor graph of the ring [Formula: see text] is a simple undirected graph with vertices representing non-zero zero-divisors in [Formula: see text]. Two vertices, denoted as a and b, are connected if there are elements x in the annihilator of a and y in the annihilator of b such that their product xy equals zero. In particular, we examine the Randić spectrum of [Formula: see text] for specific values of [Formula: see text], which are products of prime numbers and their powers.

Sandpile groups are a subtle graph isomorphism invariant, in the form of a finite abelian group, whose cardinality is the number of spanning trees in the graph. We study their group structure for graphs obtained by attaching a cone vertex to a tree. For example, it is shown that the number of generators of the sandpile group is at most one less than the number of leaves in the tree. For trees on a fixed number of vertices, the paths and stars are shown to provide extreme behavior, not only for the number of generators, but also for the number of spanning trees, and for Tutte polynomial evaluations that count the recurrent sandpile configurations by their numbers of chips. [ABSTRACT FROM AUTHOR]

A combinatorial approach is given to compute bases for eigenspaces of zero-divisor graphs of finite Boolean rings. A commutative monoid $ \mathcal {S} $ S of graphs is shown to contain a cyclic submonoid $ \mathcal {U} $ U that determines values of the entries of basis elements, while the members of its complement $ \mathcal {S}\setminus \mathcal {U} $ S ∖ U encode the supports of these elements. Furthermore, every member of $ \mathcal {S}\setminus \mathcal {U} $ S ∖ U is associated with a Catalan-triangle number, which counts the number of basis elements whose supports are determined by the given member. This is established by using a combinatorial interpretation of Catalan-triangle numbers to produce linearly independent sets of eigenvectors. [ABSTRACT FROM AUTHOR]

Let Γ be a simple finite graph with vertex set V (Γ) and edge set E (Γ) . Let R be an equivalence relation on V (Γ) . The R -super Γ graph Γ R is a simple graph with vertex set V (Γ) and two distinct vertices are adjacent if either they are in the same R -equivalence class or there are elements in their respective R -equivalence classes that are adjacent in the original graph Γ . We first show that Γ R is a generalized join of some complete graphs and using this we obtain the adjacency and Laplacian spectrum of conjugacy super commuting graphs and order super commuting graphs of dihedral group D 2 n (n ≥ 3) , generalized quaternion group Q 4 m (m ≥ 2) and the nonabelian group Z p ⋊ Z q of order pq, where p and q are distinct primes with q | (p - 1) . [ABSTRACT FROM AUTHOR]

Let 푅 be a commutative ring with identity 1 ≠ 0 and let Z ⁢ (R) ′ be the set of all non-zero and non-unit elements of ring 푅. Further, Γ ′ ⁢ (R) denotes the cozero-divisor graph of 푅, is an undirected graph with vertex set Z ⁢ (R) ′ , and w ∉ z ⁢ R and z ∉ w ⁢ R if and only if two distinct vertices 푤 and 푧 are adjacent, where q ⁢ R is the ideal generated by the element 푞 in 푅. In this paper, we find the signless Laplacian eigenvalues of the graphs Γ ′ ⁢ (Z n) for n = p 1 N ⁢ p 2 ⁢ p 3 and p 1 N ⁢ p 2 M ⁢ p 3 , where p 1 , p 2 , p 3 are distinct primes and N , M are positive integers. We also show that the cozero-divisor graph Γ ′ ⁢ (Z p 1 ⁢ p 2 ) is a signless Laplacian integral. [ABSTRACT FROM AUTHOR]

Let $ \mathcal {G}_R $ G R denote a unitary one-matching bi-Cayley graph over a finite commutative ring R. In this paper, we give a necessary and sufficient condition for $ \mathcal {G}_R $ G R (respectively, the complement of $ \mathcal {G}_R $ G R and the line graph of $ \mathcal {G}_R $ G R ) to be Ramanujan. [ABSTRACT FROM AUTHOR]

In this note, we use eigenvalue interlacing to derive an inequality between a graph’s maximum degree and its maximum and minimum adjacency eigenvalues. The equality case is fully characterized.

Let R be a finite commutative ring with unity, and let G = (V , E) be a simple graph. The zero-divisor graph, denoted by Γ (R) is a simple graph with vertex set as R, and two vertices x , y ∈ R are adjacent in Γ (R) if and only if x y = 0 . In [5], the authors have studied the Laplacian eigenvalues of the graph Γ (Z n) and for distinct proper divisors d 1 , d 2 , ⋯ , d k of n, they defined the sets as, A d i = { x ∈ Z n : (x , n) = d i } , where (x, n) denotes the greatest common divisor of x and n. In this paper, we show that the sets A d i , 1 ≤ i ≤ k are actually orbits of the group action: A u t (Γ (R)) × R ⟶ R , where A u t (Γ (R)) denotes the automorphism group of Γ (R) . Our main objective is to determine new classes of threshold graphs, since these graphs play an important role in several applied areas. For a reduced ring R, we prove that Γ (R) is a connected threshold graph if and only if R ≅ F q or R ≅ F 2 × F q . We provide classes of threshold graphs realized by some classes of local rings. Finally, we characterize all finite commutative rings with unity of which zero-divisor graphs are not threshold. [ABSTRACT FROM AUTHOR]

Let Z (R) ′ be the set of all non-unit and non-zero elements of ring R , a commutative ring with identity 1 ≠ 0 . The cozero-divisor graph of R , denoted by the notation Γ ′ (R) , is an undirected graph with vertex set Z (R) ′ . Any two distinct vertices w and z are adjacent if and only if w ∉ z R and z ∉ w R , where q R is the ideal generated by the element q in R . In this article, we evaluate the Sombor index of the graphs Γ ′ (Z n) for different values of n. Additionally, we compute Γ ′ (Z n) , the cozero-divisor graph Sombor spectrum. [ABSTRACT FROM AUTHOR]

For a commutative ring R, the comaximal graph Γ (R) of R is a simple graph with vertex set R and two distinct vertices u and v of Γ (R) are adjacent if and only if a R + b R = R . In this article, we find the Laplacian eigenvalues of Γ (Z n) and show that the algebraic connectivity of Γ (Z n) is always an even integer and equals ϕ (n) , thereby giving a large family of graphs with integral algebraic connectivity. Further, we prove that the second largest Laplacian eigenvalue of Γ (Z n) is an integer if and only if n = p α q β , and hence Γ (Z n) is Laplacian integral if and only if n = p α q β , where p, q are primes and α , β are non-negative integers. This answers a problem posed by [Banerjee, Laplacian spectra of comaximal graphs of the ring Z n , Special Matrices, (2022)]. [ABSTRACT FROM AUTHOR]

Let Γ be a finite simple graph with adjacency matrix A. Γ is said to be singular if and only if A has an eigenvalue 0. The nullity of Γ, denoted by null $ (\Gamma) $ (Γ) , is the multiplicity of the eigenvalue 0 in the spectrum of Γ. In this paper, we completely determine the singularity of graphs Γ for which the dicylic or semi-dihedral group acts transitively and faithfully on V as a group of automorphisms. Moreover, we give formulas to calculate the nullities for such graphs. [ABSTRACT FROM AUTHOR]

The fact that the adjacency matrix of every finite graph is diagonalizable plays a fundamental role in spectral graph theory. Since this fact does not hold in general for digraphs, it is natural to ask whether it holds for digraphs with certain level of symmetry. Interest in this question dates back to the early 1980 s, when P. J. Cameron asked for the existence of arc-transitive digraphs with non-diagonalizable adjacency matrix. This was answered in the affirmative by Babai (J Graph Theory 9:363–370, 1985). Then Babai posed the open problems of constructing a 2-arc-transitive digraph and a vertex-primitive digraph whose adjacency matrices are not diagonalizable. In this paper, we solve Babai's problems by constructing an infinite family of s-arc-transitive digraphs for each integer s ≥ 2 , and an infinite family of vertex-primitive digraphs, both of whose adjacency matrices are non-diagonalizable. [ABSTRACT FROM AUTHOR]

In this paper, we study the prime order Cayley graph assigned to the group ℤn for different values of n. We specify some of the graph theoretical properties such as chromatic and perfect matching numbers. Furthermore, we determine the adjacency matrices and eigenvalues of the prime order Cayley graph associated with groups ℤn and 𝒟2n.

Let F q be a finite field of q elements, V an n-dimensional vector space over F q . Recently, Wang et al. in [Automorphisms and domination numbers of transformation graphs over vector spaces. Linear Multilinear Algebra. 2019;67:1350–1363] defined a graph Γ ′ (V) with vertex set T ∪ W , where T is the set of all nonzero singular linear transformations over V and W is the set of all nontrivial subspaces of V , and there is an undirected edge between A ∈ T and W ∈ W if and only if A maps W to the zero space, that is A (W) = { 0 }. In the end of Wang et al. [Automorphisms and domination numbers of transformation graphs over vector spaces. Linear Multilinear Algebra. 2019;67:1350–1363], the authors posed an open problem: Characterizing the domination number and the automorphisms of Γ ′ (V). In the present article, we solve this problem completely. [ABSTRACT FROM AUTHOR]

For a group G, the generating graph Γ (G) is defined as the graph with the vertex set G, and any two distinct vertices of Γ (G) are adjacent if they generate G. In this paper, we study the generating graph of D n , where D n is a Dihedral group of order 2n. We explore various graph theoretic properties, and determine complete spectrum of the adjacency and the Laplacian matrix of Γ (D n) . Moreover, we compute some distance and degree based topological indices of Γ (D n) . [ABSTRACT FROM AUTHOR]

We consider a Leibniz algebra L = I ⊕ V over an arbitrary base field F , being I the ideal generated by the products [ x , x ] , x ∈ L . This ideal has a fundamental role in the study presented in our paper. A basis B = { v i } i ∈ I of L is called multiplicative if for any i , j ∈ I we have that [ v i , v j ] ∈ F v k for some k ∈ I. We associate an adequate graph Γ (L , B) to L relative to B . By arguing on this graph we show that L decomposes as a direct sum of ideals, each one being associated to one connected component of Γ (L , B). Also the minimality of L and the division property of L are characterized in terms of the weak symmetry of the defined subgraphs Γ (L , B I ) and Γ (L , B V ). [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with all non-zero zero divisors Z ∗ (R) of R . Then Γ (R) is said to be a zero divisor graph if and only if a · b = 0 where a , b ∈ V (Γ (R)) = Z ∗ (R) and (a , b) ∈ E (Γ (R)) . Graph energy E (Γ (R)) is defined as the sum of the absolute eigenvalues of the adjacency matrix of Γ (R) , then E (Γ (R)) = ∑ i = 1 n | λ i | . A topological index is a numeric quantity associated with a chemical structure that attempts to link the chemical structure to various physicochemical properties, chemical reactivity, or biological activity. This paper discusses the graph energy and various topological indices of zero divisor graph associated with the commutative ring. [ABSTRACT FROM AUTHOR]

For a finite group G, the power graph P (G) is a simple connected graph whose vertex set is the set of elements of G and two distinct vertices x and y are adjacent if and only if xi = y or yj = x, for 2 ≤ i, j ≤ n. In this paper, we obtain the distance Laplacian spectrum of power graphs of finite groups such as cyclic groups, dihedral groups, dicyclic groups, abelian groups and elementary abelian p groups. Moreover, we find the largest and second smallest distance Laplacian eigenvalue of power graphs of such groups. [ABSTRACT FROM AUTHOR]

AbstractLet R be a ring with unity. The cozero-divisor graph of a ring R, denoted by [Formula: see text] is an undirected simple graph whose vertices are the set of all non-zero and non-unit elements of R, and two distinct vertices x and y are adjacent if and only if [Formula: see text] and [Formula: see text] In this paper, first we study the Laplacian spectrum of [Formula: see text] We show that the graph [Formula: see text] is Laplacian integral. Further, we obtain the Laplacian spectrum of [Formula: see text] for [Formula: see text] where [Formula: see text] and p, q are distinct primes. In order to study the Laplacian spectral radius and algebraic connectivity of [Formula: see text] we characterized the values of n for which the Laplacian spectral radius is equal to the order of [Formula: see text] Moreover, the values of n for which the algebraic connectivity and vertex connectivity of [Formula: see text] coincide are also described. At the final part of this paper, we obtain the Wiener index of [Formula: see text] for arbitrary n.

In this paper, we characterize when t-type one-matching (respectively, homogeneous) bi-Cayley graphs over finite cyclic groups are integral. We also give a necessary and sufficient condition for some integral t-type one-matching (respectively, homogeneous) bi-Cayley graphs over finite cyclic groups of prime power order to be Ramanujan. [ABSTRACT FROM AUTHOR]

We define GG-cospectrality of two GG-gain graphs (Γ,ψ)\left(\Gamma ,\psi ) and (Γ′,ψ′)\left(\Gamma ^{\prime} ,\psi ^{\prime} ), proving that it is a switching isomorphism invariant. When GG is a finite group, we prove that GG-cospectrality is equivalent to cospectrality with respect to all unitary representations of GG. Moreover, we show that two connected gain graphs are switching equivalent if and only if the gains of their closed walks centered at an arbitrary vertex vv can be simultaneously conjugated. In particular, the number of switching equivalence classes on an underlying graph Γ\Gamma with nn vertices and mm edges, is equal to the number of simultaneous conjugacy classes of the group Gm−n+1{G}^{m-n+1}. We provide examples of GG-cospectral switching nonisomorphic graphs and we prove that any gain graph on a cycle is determined by its GG-spectrum. Moreover, we show that when GG is a finite cyclic group, the cospectrality with respect to a faithful irreducible representation implies the cospectrality with respect to any other faithful irreducible representation, and that the same assertion is false in general.

In this paper, we study the interplay between the structural and spectral properties of the comaximal graph Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) of the ring Zn{{\mathbb{Z}}}_{n} for n>2n\gt 2. We first determine the structure of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) and deduce some of its properties. We then use the structure of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) to deduce the Laplacian eigenvalues of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) for various nn. We show that Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) is Laplacian integral for n=pαqβn={p}^{\alpha }{q}^{\beta }, where p,qp,q are primes and α,β\alpha ,\beta are non-negative integers and hence calculate the number of spanning trees of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) for n=pαqβn={p}^{\alpha }{q}^{\beta }. The algebraic and vertex connectivity of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) have been shown to be equal for all nn. An upper bound on the second largest Laplacian eigenvalue of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) has been obtained, and a necessary and sufficient condition for its equality has also been determined. Finally, we discuss the multiplicity of the Laplacian spectral radius and the multiplicity of the algebraic connectivity of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}). We then investigate some properties and vertex connectivity of an induced subgraph of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}). Some problems have been discussed at the end of this paper for further research.

In this paper, the integral Cayley graphs over the nonabelian group V8n = 〈a, b | a2n = b4 = 1, ba = a−1b−1, b−1a = a−1b〉 are studied, where n is an odd integer. Character tables of V8n are used to provide a necessary and sufficient condition for the integrality of such Cayley graphs. This is then translated into a simple necessary and sufficient condition for integrality in terms of the associated Boolean algebra of 〈a2〉. This leads to a construction of several infinite families of connected integral Cayley graphs over V8n. Finally, a necessary and sufficient condition for the integrality of Cayley graphs over V8p for a prime p is given. [ABSTRACT FROM AUTHOR]

Let F q be a finite field with q elements, n ≥ 2 a positive integer, V 0 a n-dimensional vector space over F q and T 0 the set of all linear functionals from V 0 to F q . Let V = V 0 ∖ { 0 } and T = T 0 ∖ { 0 }. The linear functional graph of V 0 dented by ϝ (V) , is an undirected bipartite graph, whose vertex set V is partitioned into two sets as V = V ∪ T and two vertices v ∈ V and f ∈ T are adjacent if and only if f sends v to the zero element of F q (i.e. f (v) = 0). In this paper, the structure of all automorphisms of this graph is characterized and formalized. Also the cardinal number of automorphisms group for this graph is determined. [ABSTRACT FROM AUTHOR]

In this paper, we first study zero divisor graphs over finite chain rings. We determine their rank, determinant, and eigenvalues using reduction graphs. Moreover, we extend the work to zero divisor graphs over finite commutative principal ideal rings using a combinatorial method, finding the number of positive eigenvalues and the number of negative eigenvalues, and finding upper and lower bounds for the largest eigenvalue. Finally, we characterize all finite commutative principal ideal rings such that their zero divisor graphs are complete and compute the Wiener index of the zero divisor graphs over finite commutative principal ideal rings. [ABSTRACT FROM AUTHOR]

In this paper, we determine the energies of complements of unitary one-matching bi-Cayley graphs over finite commutative rings. We also determine the energies of some one-matching bi-gcd-graphs as well as the energies of complements of some one-matching bi-gcd-graphs, which are generalizations of unitary one-matching bi-Cayley graphs. [ABSTRACT FROM AUTHOR]

The proper power graph P ∗ (G) of a group G is the simple graph with non-trivial elements of G as vertices and two distinct elements are adjacent if and only if one is a power of the other. The aim of this paper is to find the structure and to compute the spectrum of the proper power graph of direct product of two finite groups G 1 and G 2 , where at least one of G 1 and G 2 is an EPO group. Also we provide the full spectrum and hence the energy of P ∗ (G 1 × G 2) whenever both G 1 and G 2 are EPO groups. [ABSTRACT FROM AUTHOR]

The power graph of a group Ω is a graph, whose node set is Ω and two distinct elements are adjacent if and only if one is an integral power of the other. A metric dimension of a graph Γ, denoted by ψ(Γ) is the minimum cardinality of the resolving set of Γ. In this context, we study distant properties and detour distant properties such as closure, interior, distance degree sequence and eccentric subgraphs of the power graphs of certain finite non-abelian groups. As a consequence, we figure out the metric dimension and resolving polynomial of power graphs for dihedral and generalized quaternion groups by using neighbourhood and twin sets. [ABSTRACT FROM AUTHOR]

Let R be a ring with unity which is graded by a cancellative partial groupoid (magma) S. A homogeneous element 0 ≠ x ∈ R is said to be locally right (left) invertible if there exist an idempotent element e ∈ S and x r ∈ R ( x l ∈ R ) such that x x r = 1 e (x l x = 1 e) where 1 e ≠ 0 is a unity of the ring R e. Element x is said to be locally two-sided invertible if it is both locally right and locally left invertible. The set of all locally invertible elements (left, right, two-sided) of R is denoted by U l (R). The homogeneous co-maximal graph Γ h (R) of R is defined as a graph whose vertex set consists of all homogeneous elements of R which do not belong to U l (R) , and distinct vertices x and y are adjacent if and only if x R + y R = R. If the edge set of Γ h (R) is nonempty, then S (with zero) contains a single (nonzero) idempotent element. This condition characterizes the connectedness of Γ h (R) \ { 0 } for a class of groupoid graded rings R which are graded semisimple, graded right Artinian, and which contain more than one maximal graded modular right ideal. If F q is a finite field and n ≥ 2 , then the full matrix ring M n (F q) is naturally graded by a groupoid S with a single nonzero idempotent element. We obtain various parameters of Γ h (M n (F q)) \ { 0 M n (F q) }. If R is S-graded, with the support equal to S \ { 0 } , and if Γ h (R) ≅ Γ h (M n (F q)) , then we prove that R and M n (F q) are graded isomorphic as S-graded rings. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with nonzero identity and let Γ (R) denote the zero divisor graph of R. In this paper, we describe the signless Laplacian and normalized Laplacian spectrum of the zero divisor graph Γ (Z n) , and we determine these spectrums for some values of n. We also characterize the cases that 0 is a signless Laplacian eigenvalue of Γ (Z n) . Moreover, we find some bounds for some eigenvalues of the signless Laplacian and normalized Laplacian matrices of Γ (Z n) . [ABSTRACT FROM AUTHOR]

For a finite commutative ring R with unity, the looped zero divisor graph Γ ˚ (R) is an undirected graph with all non-zero zero divisors of R as vertices and two vertices (not necessarily distinct) u and v are adjacent if and only if u v = 0 . The universal adjacency matrix of a looped graph G ˚ is U (G ˚) = α A + β D + γ I + η J , where α (≠ 0) , β , γ , η ∈ R , I is the identity matrix, J is the all one matrix, A is the adjacency and D is the degree diagonal matrix of G ˚ . For every non-prime integer n with ξ number of proper divisors, we show that ξ eigenpairs of U (Γ ˚ (Z n)) can be obtained from a symmetric matrix W U of order ξ , and determine explicitly all the remaining eigenpairs of U (Γ ˚ (Z n)) . As a consequence, we also study the adjacency, Seidel, Laplacian and signless Laplacian spectra of Γ ˚ (Z n) . Moreover, for n = p m , a prime p and integer m ≥ 2 , we determine the characteristic polynomial of corresponding W U matrix (except for η + α ≠ 0 with η ≠ 0 ). [ABSTRACT FROM AUTHOR]

We prove the alternating sign conjecture for the perfect matching derangement graph. [ABSTRACT FROM AUTHOR]

Let G be a finite group and Γ be a (di)graph. Then Γ is called an n-Cayley (di)graph over G if Aut (Γ) admits a semiregular subgroup isomorphic to G with n orbits on V (Γ). In this paper, we determine the normalized Laplacian polynomial of n-Cayley (di)graphs over a group G in terms of irreducible representations of G. We give exact formulas for the normalized Laplacian eigenvalues of 2-Cayley graphs over abelian groups. Among other results, as an application, we prove that the degree-Kirchhoff index of the n-sunlet graph is n (2 n − 1) (2 n + 7) 3 . [ABSTRACT FROM AUTHOR]

Topological indices are widely used in mathematical chemistry to study graphs of molecules of a chemical compound. The comaximal graph of a ring R, denoted by Γ (R) , has elements of R as vertices, and any two distinct vertices x, y of Γ (R) are adjacent if and only if R x + R y = R . In this paper, we find the eigenvalues of the adjacency matrix, and the topological indices of the comaximal graph Γ (Z n) of the ring Z n . We determine the Wiener index, the first and the second Zagreb index, as well as the adjacency and the Laplacian energy of Γ (Z n) . A SAGE code for evaluating the above indices has also been provided at the end of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the dihedral groups and the dicyclic groups, and consider their corresponding integral Cayley graphs. We obtain the sufficient conditions for the integrality of the distance powers Γ D of the Cayley graph Γ = X (D 2 n , S) (resp. Γ = X (T 4 n , S)) ( n ≥ 3) for a set of nonnegative integers D. In particular, for a prime p, we show that if Γ = X (D 2 p , S) (resp. Γ = X (T 4 p , S)) is integral, then the distance powers of Γ = X (D 2 p , S) (resp. Γ = X (T 4 p , S)) are integral Cayley graphs. [ABSTRACT FROM AUTHOR]

Let G be a finite abelian group viewed a Z -module and let G = (V , E) be a simple graph. In this paper, we consider a graph Γ (G) called as a group-annihilator graph. The vertices of Γ (G) are all elements of G and two distinct vertices x and y are adjacent in Γ (G) if and only if [ x : G ] [ y : G ] G = { 0 } , where x , y ∈ G and [ x : G ] = { r ∈ Z : r G ⊆ Z x } is an ideal of a ring Z . We discuss in detail the graph structure realised by a group G. Moreover, we study the creation sequence, hyperenergeticity and hypoenergeticity of group-annihilator graphs. Finally, we conclude the paper with a discussion on Laplacian eigen values of the group-annihilator graph. We show that the Laplacian eigen values are representatives of orbits of the group action: A u t (Γ (G)) × G → G . [ABSTRACT FROM AUTHOR]

We investigate the abelian sandpile group on modified wheels Wˆn{\hat{W}}_{n} by using a variant of the dollar game as described in [N. L. Biggs, Chip-Firing and the critical group of a graph, J. Algebr. Comb. 9 (1999), 25–45]. The complete structure of the sandpile group on a class of graphs is given in this paper. In particular, it is shown that the sandpile group on Wˆn{\hat{W}}_{n} is a direct product of two cyclic subgroups generated by some special configurations. More precisely, the sandpile group on Wˆn{\hat{W}}_{n} is the direct product of two cyclic subgroups of order an{a}_{n} and 3an3{a}_{n} for n even and of order an{a}_{n} and 2an2{a}_{n} for n odd, respectively.

Let H be a group and Y be a non-empty subset of H. The commuting graph Γ = C (H , Y) has Y as the node set, where y 1 , y 2 ∈ Y are adjacent whenever y 1 and y 2 commute in H. In this paper, we discuss the node connectivity, the edge connectivity, and the planarity of commuting graphs of finite cyclic groups, dihedral and dicyclic groups. We prove that these graphs are not Hamiltonian. By studying the characteristic polynomials, we obtain bounds of the spectral radii of their commuting graphs. [ABSTRACT FROM AUTHOR]

Let Δ be a digraph of diameter 2 with the maximum undirected vertex degree t and the maximum directed out-degree z. The largest possible number v of vertices of Δ is given by the following generalization of the Moore bound: v ≤ (z + t) 2 + z + 1 , and a digraph attaining this bound is called a Moore digraph. Apart from the case t = 1 , only three Moore digraphs are known, which are also Cayley graphs. Using computer search, Erskine (J Interconnect Netw, 17: 1741010, 2017) ruled out the existence of further examples of Cayley digraphs attaining the Moore bound for all orders up to 485. We use an algebraic approach to this problem, which goes back to an idea of G. Higman from the theory of association schemes, also known as Benson's Lemma in finite geometry, and show non-existence of Moore Cayley digraphs of certain orders. [ABSTRACT FROM AUTHOR]

We extend the notion of the compressed zero-divisor graph Θ (R) to noncommutative rings in a way that still induces a product preserving functor Θ from the category of finite unital rings to the category of directed graphs. For a finite field F, we investigate the properties of Θ (M n (F)) , the graph of the matrix ring over F, and give a purely graph-theoretic characterization of this graph when n ≠ 3. For n ≠ 2 we prove that every graph automorphism of Θ (M n (F)) is induced by a ring automorphism of M n (F). We also show that for finite unital rings R and S, where S is semisimple and has no homomorphic image isomorphic to a field, if Θ (R) ≅ Θ (S) , then R ≅ S. In particular, this holds if S = M n (F) with n ≠ 1. [ABSTRACT FROM AUTHOR]

The reduced power graph of a group G is the graph whose vertices are the elements of G and two vertices u and v are adjacent if and only if u ⊂ v or v ⊂ u . In this paper, we study the Laplacian spectrum of the reduced power graph of the additive cyclic group Z n , some finite abelian p-groups, dihedral group D 2 n , quaternion group Q 4 n and semi-dihedral group S D 8 n . [ABSTRACT FROM AUTHOR]

We study the balance of G-gain graphs, where G is an arbitrary group, by investigating their adjacency matrices and their spectra. As a first step, we characterize switching equivalence and balance of gain graphs in terms of their adjacency matrices in M n (C G) . Then we introduce a represented adjacency matrix, associated with a gain graph and a group representation, by extending the theory of Fourier transforms from the group algebra C G to the algebra M n (C G) . We prove that, anytime G admits a finite-dimensional faithful unitary representation π , a G-gain graph is balanced if and only if the spectrum of the represented adjacency matrix associated with π coincides with the spectrum of the underlying graph, with multiplicity given by the degree of the representation. We show that the complex adjacency matrix of complex unit gain graphs and the adjacency matrix of a cover graph are indeed particular cases of our construction. This enables us to recover some classical results and prove some new characterizations of balance in terms of spectrum, index or structure of these graphs. [ABSTRACT FROM AUTHOR]