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The eccentricity matrix ℰ (G) = ( u v) of a graph G is constructed from the distance matrix by keeping each row and each column only the largest distances with u v = d (u , v) , if d (u , v) = min { (u) , (v) } , 0 , otherwise , where d (u , v) is the distance between two vertices u and v , and (u) = max { d (u , v) | v ∈ V (G) } is the eccentricity of the vertex u. The ℰ -eigenvalues of G are those of its eccentricity matrix. In this paper, employing the well-known Cauchy Interlacing Theorem we give the following lower bounds for the second, the third and the fourth largest ℰ -eigenvalues by means of the diameter d of G : ξ 2 (G) ≥ − 1 , if d ≤ 2 ; α d , if d ≥ 3 , ξ 3 (G) ≥ − d , and ξ 4 (G) ≥ − 1 − 5 2 d , where α = 0. 3 1 1 1 + is the second largest root of x 3 − x 2 − 3 x + 1 = 0. Moreover, we further discuss the graphs achieving the above lower bounds. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the traces of the adjacency tensor of uniform hypergraphs (simply called the traces of hypergraphs). We give new expressions for the traces of hypertrees and linear unicyclic hypergraphs by the weight function assigned to their connected sub-hypergraphs, and present some perturbation results for the traces of a hypergraph when its structure is locally changed. As an application we determine the unique hypertree with maximum Estrada index among all hypertrees with perfect matchings and given number of edges, and the unique unicyclic hypergraph with maximum Estrada index among all unicyclic hypergraph with girth 3 and given number of edges. [ABSTRACT FROM AUTHOR]

The distance energy of a connected graph G is the sum of absolute values of the eigenvalues of the distance matrix of G. In this paper, we study how the distance energy of the complete split graph G S (m , n) = K m + K ¯ n changes when an edge is deleted from it. The complete split graph G S (m , n) consists of a clique on m vertices and an independent set on n vertices in which each vertex of the clique is adjacent to each vertex of the independent set. We prove that the distance energy of the complete split graph G S (m , n) always increases when an edge is deleted from it. [ABSTRACT FROM AUTHOR]

Let ℤn be the ring of integer modulo n with two binary operators, addition (+) and multiplication (.), where n is a positive integer. The special set 풮 is defined as 풮 = {a ∈ ℤn : (∃b ∈ ℤn) ba ≡ a, b≢a, b≢1}. Our purpose in the present paper is to propose a new family of interconnection networks that are Cayley graphs on this special set 풮 and denote it by Ω−(Z n). In this paper, we define a relationship between G and Ge∗, Ge∗ is a derived graph from G by removing r edges, where r is a known fixed value. We also give the spectrum of absorption Cayley graph, unitary addition Cayley graph, and Ω−(Z n). We also provide values of n for which the graph Ω−(ℤ n) is hyperenergetic and discuss the structural properties of this graph, such as planarity and connectedness. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

If G is a simple graph on n vertices v 1 , v 2 , ... , v n and d i be the degree of i th vertex v i then the average degree matrix of graph G , AD (G) is of order n × n whose (i , j) th entry is d i + d j 2 if the vertices v i and v j are adjacent and zero otherwise. The average degree energy of G , AD (E (G)) is the sum of all absolute value of eigenvalues of average degree matrix of a graph G. In this paper, bounds for average degree energy AD (E (G)) and the relation between average degree energy AD (E (G)) and energy E (G) is discussed. [ABSTRACT FROM AUTHOR]

Let G be an orientation of a simple graph G with n vertices and m edges. The skew Laplacian matrix SL (G) of the digraph G is defined as SL (G) = D ̃ (G) − i S (G) , where i = − 1 is the imaginary unit, D ̃ (G) is the diagonal matrix with oriented degrees α i = d i + − d i − as diagonal entries and S (G) is the skew matrix of the digraph G. The largest eigenvalue of the matrix SL (G) is called skew Laplacian spectral radius of the digraph G. In this paper, we study the skew Laplacian spectral radius of the digraph G. We obtain some sharp lower and upper bounds for the skew Laplacian spectral radius of a digraph G , in terms of different structural parameters of the digraph and the underlying graph. We characterize the extremal digraphs attaining these bounds in some cases. Further, we end the paper with some problems for the future research in this direction. [ABSTRACT FROM AUTHOR]

In this paper, we prove that every simple connected graph with α > n 2 has 1 as a normalized Laplacian eigenvalue with multiplicity at least 2 α − n , where n and α are the order and the independence number of the graph, respectively. Then we investigate graphs with α ≤ n 2 and having or not having 1 as a normalized Laplacian eigenvalue. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with unity, and let Γ (R) denote the comaximal graph of R. The comaximal graph Γ (R) has vertex set as R, and any two distinct vertices x, y of Γ (R) are adjacent if R x + R y = R. Let Γ 2 (R) denote the induced subgraph of Γ (R) on the set of all nonzero non-unit elements of R, and any two distinct vertices x, y of Γ 2 (R) are adjacent if R x + R y = R. In this paper, we study the graphical structure as well the adjacency spectrum of Γ 2 (ℤ n) , where n ≥ 4 is a non-prime positive integer, and ℤ n is the ring of integers modulo n. We show that for a given non-prime positive integer n with D number of positive proper divisors, the eigenvalues of Γ 2 (ℤ n) are 0 with multiplicity n − φ (n) − D − 1 , and remaining eigenvalues are contained in the spectrum of a symmetric D × D matrix. We further calculate the rank and nullity of Γ 2 (ℤ n). We also determine all the eigenvalues of Γ 2 (ℤ n) whenever Γ 2 (ℤ n) is a bipartite graph. Finally, apart from determining certain structural properties of Γ 2 (ℤ n) , we conclude the paper by determining the metric dimension of Γ 2 (ℤ n). [ABSTRACT FROM AUTHOR]

Let n ≥ 1 and Un be the multiplicative group of the additive group ℤn of integers modulo n. The unitary Haar graph Hn = Haar(ℤn,Un) is a graph with vertex set ℤn ×{0, 1} and the edge set {{(a, 0), (b, 1)}|b − a ∈ Un}. In this paper, we study some structural properties of Hn such as total chromatic number, diameter, planarity, the girth, vertex-connectivity and edge-connectivity. Also, some spectral properties of the graph Hn are determined. [ABSTRACT FROM AUTHOR]

For a d -regular graph G , an edge-signing σ : E (G) → { − 1 , 1 } is called a good signing if the absolute eigenvalues of its adjacency matrix are at most 2 d − 1 . Bilu and Linial conjectured that for each regular graph there exists a good signing. In this paper, by using the concept of "Equitable Partition", we prove the conjecture for some cases. We show how to find out a good signing for special complete graphs and lexicographic product of two graphs. In particular, if there exist two good signings for graph G , then we can find a good signing for a 2-lift of G. [ABSTRACT FROM AUTHOR]

The Seidel spectrum of a graph is defined as the multiset of all eigenvalues of its Seidel matrix. For two simple connected graphs G 1 and G 2 , let us denote the subdivision-vertex join and subdivision-edge join by G 1 ∨ ̇ G 2 and G 1 ⊻ G 2 , respectively. In this paper, we completely determine the Seidel spectrum and corresponding Seidel eigenvectors of G 1 ∨ ̇ G 2 and G 1 ⊻ G 2 . As an application, we give a sufficient and necessary condition for G 1 ∨ ̇ G 2 and G 1 ⊻ G 2 to be Seidel integral. [ABSTRACT FROM AUTHOR]

A conjugate skew gain graph is a skew gain graph with the labels (also called, the conjugate skew gains) from the field of complex numbers on the oriented edges such that they get conjugated when we reverse the orientation. In this paper, we introduce distance matrices for conjugate skew gain graphs and characterize balanced conjugate skew gain graphs using these matrices. We provide explicit formulae for the distance spectra of certain conjugate skew gain graphs. [ABSTRACT FROM AUTHOR]

For a simple connected graph G of order n, the normalized Laplacian is a square matrix of order n, defined as ℒ (G) = D (G) − 1 2 L (G) D (G) − 1 2 , where D (G) − 1 2 is the diagonal matrix whose i-th diagonal entry is 1 d i . In this paper, we find the normalized Laplacian eigenvalues of the joined union of regular graphs in terms of the adjacency eigenvalues and the eigenvalues of quotient matrix associated with graph G. For a finite group , the power graph () of a group is defined as the simple graph in which two distinct vertices are joined by an edge if and only if one is the power of the other. As a consequence of the joined union of graphs, we investigate the normalized Laplacian eigenvalues of the power graphs of the finite cyclic group ℤ n . [ABSTRACT FROM AUTHOR]

The rotational dimension is a minor-monotone graph invariant related to the dimension of a Euclidean space containing a spectral embedding corresponding to the first nonzero eigenvalue of the graph Laplacian, which is introduced by Göring, Helmberg and Wappler. In this paper, we study rotational dimensions of graphs which contain large complete graphs. The complete graph is characterized by its rotational dimension. It will be obtained that a chordal graph may be made large while keeping the rotational dimension constant. [ABSTRACT FROM AUTHOR]

The Estrada index is a graph-spectrum-based invariant having an important role in chemistry and physics. In this paper, we define the generalized adjacency Estrada index of a graph and obtain some lower and upper bounds for the generalized adjacency Estrada index in terms of various graph parameters associated with the structure of a graph. Further, we characterize the extremal graphs attaining these bounds. We also highlight the relationship between generalized adjacency Estrada index and generalized adjacency energy. Using these results, we obtain the improved bounds for the Estrada index based on the adjacency eigenvalues of a graph. [ABSTRACT FROM AUTHOR]

Let G be a connected graph of order n and let RD (G) be the reciprocal distance matrix (also called Harary matrix) of the graph G. Let ρ 1 ≥ ρ 2 ≥ ⋯ ≥ ρ n be the eigenvalues of the reciprocal distance matrix RD (G) of the connected graph G called the reciprocal distance eigenvalues of G. The Harary energy HE (G) of a connected graph G is defined as sum of the absolute values of the reciprocal distance eigenvalues of G , that is, HE (G) = ∑ i = 1 n | ρ i |. In this paper, we establish some new lower and upper bounds for HE (G) , in terms of different graph parameters associated with the structure of the graph G. We characterize the extremal graphs attaining these bounds. We also obtain a relation between the Harary energy and the sum of k largest adjacency eigenvalues of a connected graph. [ABSTRACT FROM AUTHOR]

Let G be a graph of which A be the adjacency matrix and D be a diagonal matrix whose diagonal entries are the degrees of vertices of G. Q = A + D is known as the signless Laplacian matrix of the graph G. For any real α ∈ [ 0 , 1 ] , A α matrix is defined by the convex combination α D + (1 − α) A of D and A. Clearly, A α coincides with A for α = 0 and coincides with Q / 2 for α = 1 / 2. Thus, the A α eigenvalues are generalizations of adjacency eigenvalues and signless Laplacian eigenvalues. In this paper, we have obtained the A α spectra of corona and edge corona of two graphs. We have also shown that A α cospectral graphs and α equienergetic graphs can be obtained from there. [ABSTRACT FROM AUTHOR]

Let D be a digraph of order n and let A (D) be the adjacency matrix of D. Let Deg (D) be the diagonal matrix of vertex out-degrees of D. For any real α ∈ [ 0 , 1 ] the generalized adjacency matrix A α (D) of the digraph D is defined as A α (D) = α Deg (D) + (1 − α) A (D). This matrix generalizes the spectral theories of the adjacency matrix and the signless Laplacian matrix of D. In this paper, we find A α -spectrum of the joined union of digraphs in terms of spectrum of adjacency matrices of its components and the eigenvalues of an auxiliary matrix determined by the joined union. We determine the A α -spectrum of join of two regular digraphs and the join of a regular digraph with the union of two regular digraphs of distinct degrees. As applications, we obtain the A α -spectrum of various families of unsymmetric digraphs. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new framework that's we call the weighted 3 -simplicial complex and we define its spectral gap. An upper bound for our spectral gap is given by generalizing the Cheeger constant. The lower bound for our spectral gap is obtained from the first nonzero eigenvalue of the Laplacian acting on the functions of certain weighted 2 -simplicial complexes. [ABSTRACT FROM AUTHOR]

Let G be a graph with n vertices and m edges. The minimum edge dominating energy is equal to the sum of the absolute values of eigenvalues of the minimum edge dominating matrix of graph G. In this paper, we establish some bounds for the minimum edge dominating energy of subgraphs of a graph. [ABSTRACT FROM AUTHOR]

Given a graph G with vertex set V (G) = { v 1 , v 2 , ... , v n } , we associate to it a P (k) path matrix whose (i , j) entry represents the maximum number of vertex disjoint path with the length k from v i to v j for i ≠ j and P i j (k) = 0 if i = j. The eigenvalues of this matrix are called the path eigenvalues of the graph. In this note, we investigate P (k) path energy of some classes of graphs and several results concerning P (k) path energy has been obtained. [ABSTRACT FROM AUTHOR]

Let G be a simple undirected graph. Three new corona products of graphs based on splitting graph of G are defined. The adjacency spectra of the three new graphs based on splitting graph of G are determined. The number of spanning trees and the Kirchoff index of the new graphs are determined using their nonzero Laplacian eigenvalues. [ABSTRACT FROM AUTHOR]

We define the path Laplacian matrix and the path signless Laplacian matrix of a simple connected graph G as P ℒ (G) = T r P (G) − P (G) and P (G) = T r P (G) + P (G) , respectively, where P (G) is the path matrix and T r P (G) is the diagonal matrix of the vertex transmissions. The generalized path matrix is P α (G) = α T r P (G) + (1 − α) P (G) , for 0 ≤ α ≤ 1 and ρ 1 α ≥ ρ 2 α ≥ ⋯ ≥ ρ n α are the eigenvalues of P α (G). The generalized path energy can be expressed as E P α (G) = ∑ i = 1 n ρ i α − 2 α P W (G) n , where P W (G) is the path Wiener index of G. We give basic properties of generalized path matrix P α (G). Also, some upper and lower bounds of the generalized path energy of some graphs are studied. [ABSTRACT FROM AUTHOR]