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Let S⊆C*=C{0} and A∈Mn(C). The matrix A is called an S-GHMn if A∈Mn(S) and AA*=Diag(λ1,... λn), for some positive numbers λi, i=1,... n. In this paper we provide some necessary conditions on n for the existence of an S-GHMn over a finite set S. We conjecture that for every positive integer n, there exists a {±1, ±2, ±3}-GHMn  

The zero-divisor graph of a ring R is defined as the directed graph Γ (R) that its vertices are all non-zero zero-divisors of R in which for any two distinct vertices x and y, x → y is an edge if and only if x y = 0. Recently, it has been shown that for any finite ring R, Γ (R) has an even number of edges. Here we give a simple proof for this result. In this paper we investigate some properties of zero-divisor graphs of matrix rings and group rings. Among other results, we prove that for any two finite commutative rings R, S with identity and n, m ≥ 2, if Γ (Mn (R)) ≃ Γ (Mm (S)), then n = m, | R | = | S |, and Γ (R) ≃ Γ (S). © 2007 Elsevier Inc. All rights reserved  

We say that two graphs G and H with the same vertex set commute if their adjacency matrices commute. In this article, we show that for any natural number r, the complete multigraph Knr is decomposable into commuting perfect matchings if and only if n is a 2-power. Also, it is shown that the complete graph Kn is decomposable into commuting Hamilton cycles if and only if n is a prime number. © 2006 Wiley Periodicals, Inc  

Let G be a graph. A spanning subgraph of G is called a {1, 2}-factor if each of its components is a regular graph of degree one or two. In this paper we provide a short proof of a theorem of Tutte which says that a graph G has a {1, 2}-factor if and only if i(GS) ≤ |S| for any S ⊆ V(G), where i(GS) denotes the number of isolated vertices of GS  

A multicolored tree is a tree whose edges have different colors. Brualdi and Hollingsworth [5] proved in any proper edge coloring of the complete graph K2n(n > 2) with 2n -1 colors, there are two edge-disjoint multicolored spanning trees. In this paper we generalize this result showing that if (a1,..., ak) is a color distribution for the complete graph Kn, n ≥ 5, such that 2 ≤ a1 ≤ a2 ≤ ⋯ ≤ ak ≤ (n + 1)/2, then there exist two edge-disjoint multicolored spanning trees. Moreover, we prove that for any edge coloring of the complete graph Kn with the above distribution if T is a non-star multicolored spanning tree of Kn, then there exists a multicolored spanning tree T′ of Kn such that T and T′... 

In a manner analogous to the commutative case, the zero-divisor graph of a non-commutative ring R can be defined as the directed graph Γ (R) that its vertices are all non-zero zero-divisors of R in which for any two distinct vertices x and y, x → y is an edge if and only if xy = 0. We investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of Γ (R). In this paper it is shown that, with finitely many exceptions, if R is a ring and S is a finite semisimple ring which is not a field and Γ (R) ≃ Γ (S), then R ≃ S. For any finite field F and each integer n ≥ 2, we prove that if R is a ring and Γ (R) ≃ Γ (Mn), then R ≃ Mnn. Redmond defined the simple... 

Ryser conjectured that the number of transversals of a latin square of order n is congruent to n modulo 2. Balasubramanian has shown that the number of transversals of a latin square of even order is even. A 1-factor of a latin square of order n is a set of n cells no two from the same row or the same column. We prove that for any latin square of order n, the number of 1-factors with exactly n - 1 distinct symbols is even. Also we prove that if the complete graph K2n, n ≥ 8, is edge colored such that each color appears on at most n-2 2e edges, then there exists a multicolored perfect matching. © 2004 Wiley Periodicals, Inc  

Let R be a commutative ring and Γ (R) be its zero-divisor graph. In this paper it is shown that for any finite commutative ring R, the edge chromatic number of Γ (R) is equal to the maximum degree of Γ (R), unless Γ (R) is a complete graph of odd order. In [D.F. Anderson, A. Frazier, A. Lauve, P.S. Livingston, in: Lecture Notes in Pure and Appl. Math., Vol. 220, Marcel Dekker, New York, 2001, pp. 61-72] it has been proved that if R and S are finite reduced rings which are not fields, then Γ (R) ≃ Γ (S) if and only if R ≃ S. Here we generalize this result and prove that if R is a finite reduced ring which is not isomorphic to ℤ2 × ℤ 2 or to ℤ6 and S is a ring such that Γ (R) ≃ Γ (S), then R ≃... 

The energy ϵ(G) of a graph G is the sum of the absolute values of all eigenvalues of G. Zhou in (MATCH Commun. Math. Comput. Chem. 55 (2006) 91-94) studied the problem of bounding the graph energy in terms of the minimum degree together with other parameters. He proved his result for quadrangle-free graphs. Recently, in (MATCH Commun. Math. Comput. Chem. 81 (2019) 393-404) it is shown that for every graph G, ϵ(G) ≥ 2δ(G), where δ(G) is the minimum degree of G, and the equality holds if and only if G is a complete multipartite graph with equal size of parts. Here, we provide a short proof for this result. Also, we give an affirmative answer to a problem proposed in (MATCH Commun. Math.... 

The accurate measurement of surface roughness is essential in ensuring the desired quality of machined parts. The most common method of measuring the surface roughness of machined parts is using a surface profile-meter with a contact stylus, which can provide direct measurements of surface profiles. This method has its own disadvantageous such as workpiece surface damage due to mechanical contact between the stylus and the surface, In this paper we proposed a eontactless method using image processing and artificial neural network as a pattern classifier. Having trained the network for any specific workpiece with 10 sample patterns, the system would learn how to approximate the actual surface... 

Recently, an approximate boundary condition [Opt. Lett. 38, 3009 (2013)] was proposed for fast analysis of onedimensional periodic arrays of graphene ribbons by using the Fourier modal method (FMM). Correct factorization rules are applicable to this approximate boundary condition where graphene is modeled as surface conductivity. We extend this approach to obtain the optical properties of two-dimensional periodic arrays of graphene. In this work, optical absorption of graphene squares in a checkerboard pattern and graphene nanodisks in a hexagonal lattice are calculated by the proposed formalism. The achieved results are compared with the conventional FMM, in which graphene is modeled as a... 

Recently, an approximate formalism [Opt. Express 23, 2764, (2015)] called diffractive interface theory has been reported for the fast analysis of the optical response of metasurfaces, subwavelength two-dimensional periodic arrays. In this method, the electromagnetic boundary conditions are derived using the susceptibility distribution of the metasurface, such that the analysis of metasurface is possible without solving any eigenvalue equation inside the grating layer. In this paper, we modify the boundary conditions to achieve more accurate results. In addition, in this paper, correct Fourier factorization rules are also applied leading to faster convergence rate. The obtained results are... 

Let G be a graph. An edge orientation of G is called smooth if the in-degree and the out-degree of every vertex differ by at most one. In this paper, we show that if G is a 2-edge-connected non-bipartite graph with δ(G)≥3, then G has a smooth primitive orientation. Among other results, using the spectral radius of digraphs, we show that if D1 is a primitive regular orientation and D2 is a non-regular orientation of a given graph, then for sufficiently large t, the number of closed walks of length t in D1 is more than the number of closed walks of length t in D2. © 2016 Elsevier Inc  

The singular values of a matrix A are defined as the square roots of the eigenvalues of A⁎A, and the energy of A denoted by E(A) is the sum of its singular values. The energy of a graph G, E(G), is defined as the sum of absolute values of the eigenvalues of its adjacency matrix. In this paper, we prove that if A is a Hermitian matrix with the block form A=(BDD⁎C), then E(A)≥2E(D). Also, we show that if G is a graph and H is a spanning subgraph of G such that E(H) is an edge cut of G, then E(H)≤E(G), i.e., adding any number of edges to each part of a bipartite graph does not decrease its energy. Let G be a connected graph of order n and size m with the adjacency matrix A. It is well-known... 

A circular zero-sum flow for a graph G is a function f: E(G) → R { 0 } such that for every vertex v, ∑e∈Evf(e)=0, where Ev is the set of all edges incident with v. If for each edge e, 1 ≤ | f(e) | ≤ r- 1 , where r≥ 2 is a real number, then f is called a circular zero-sum r-flow. Also, if r is a positive integer and for each edge e, f(e) is an integer, then f is called a zero-sum r-flow. If G has a circular zero-sum flow, then the minimum r≥ 2 for which G has a circular zero-sum r-flow is called the circular zero-sum flow number of G and is denoted by Φ c(G). Also, the minimum integer r≥ 2 for which G has a zero-sum r-flow is called the flow number for G and is denoted by Φ (G). In this... 

For an undirected graph G, a zero-sum flow is an assignment of non-zero real numbers to the edges, such that the sum of the values of all edges incident with each vertex is zero. It has been conjectured that if a graph G has a zero-sum flow, then it has a zero-sum 6-flow. We prove this conjecture and Bouchet's Conjecture for bidirected graphs are equivalent. Among other results it is shown that if G is an r-regular graph (r ≥ 3), then G has a zero-sum 7-flow. Furthermore, if r is divisible by 3, then G has a zero-sum 5-flow. We also show a graph of order n with a zero-sum flow has a zero-sum (n + 3)2-flow. Finally, the existence of k-flows for small graphs is investigated  

A graph is called equimatchable if all of its maximal matchings have the same size. Kawarabayashi, Plummer, and Saito showed that the only connected equimatchable 3-regular graphs are K4 and K3, 3. We extend this result by showing that for an odd positive integer r, if G is a connected equimatchable r-regular graph, then G ϵ {Kr+1, Kr,r}. Also it is proved that for an even r, a connected triangle-free equimatchable r-regular graph is isomorphic to one of the graphs C5, C7, and Kr,r. © 2017 Wiley Periodicals, Inc  

Let A be a Hermitian matrix whose graph is G (i.e. there is an edge between the vertices i and j in G if and only if the (i,j) entry of A is non-zero). Let λ be an eigenvalue of A with multiplicity mA(λ). An edge e=ij is said to be Parter (resp., neutral, downer) for λ,A if mA(λ)-mA-e(λ) is negative (resp., 0, positive ), where A-e is the matrix resulting from making the (i,j) and (j,i) entries of A zero. For a tree T with adjacency matrix A a subset S of the edge set of G is called an edge star set for an eigenvalue λ of A, if |S|=mA(λ) and A-S has no eigenvalue λ. In this paper the existence of downer edges and edge star sets for non-zero eigenvalues of the adjacency matrix of a tree is... 

A colorful path in a graph G is a path with χ(G) vertices whose colors are different. A v-colorful path is such a path, starting from v. Let G ≠ C7 be a connected graph with maximum degree Δ (G). We show that there exists a (Δ(G)+1)-coloring of G with a v-colorful path for every v ∈ V (G). We also prove that this result is true if one replaces (Δ (G) + 1) colors with 2 χ(G) colors. If χ (G) = ω(G), then the result still holds for χ(G) colors. For every graph G, we show that there exists a χ(G)-coloring of G with a rainbow path of length ⌊ χ(G)/2⌋ starting from each v ∈ V (G)  

A harmonious coloring of G is a proper vertex coloring of G such that every pair of colors appears on at most one pair of adjacent vertices. The harmonious chromatic number of G, h(G), is the minimum number of colors needed for a harmonious coloring of G. We show that if T is a forest of order n with maximum degree Δ(T)≥n+23, then h(T) = {Δ(T)+2,if T has non-adjacent vertices of degree Δ(T);Δ(T)+1,otherwise.Moreover, the proof yields a polynomial-time algorithm for an optimal harmonious coloring of such a forest