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Let G = {g1,...,gn} be a finite abelian group. Consider the complete graph with the vertex set {g1.....,.....g n}. The G-coloring of Kn is a proper edge coloring in which the color of edge {gi,gj} gi g i + gj, 1 ≤ i < 3 ≤ n. We prove that in the G-coloring of the complete graph Kn, there exists a multicolored Hamilton path if G is not an elementary abelian 2-group. Furthermore, we show that if n is odd, then the G-coloring of Kn can be decomposed into multicolored 2-factors and there are exactly lr/2 multicolored r-uniform 2-factors in this decomposition where lr is the number of elements of order r in G, 3 ≤ r ≤ n. This provides a generalization of a recent result due to Constantine which... 

For every natural number h, a graph G is said to be h-magic if there exists a labelling l:E(G)→Zh{0} such that the induced vertex set labelling l+:V(G)→Zh defined byl+(v)=∑uv∈E(G)l(uv), is a constant map. When this constant is zero, it is said that G admits a zero-sum h-magic labelling. The null set of a graph G, denoted by N(G), is the set of all natural numbers h∈N such that G admits an h-zero-sum magic labelling. In 2007, E. Salehi determined the null set of complete bipartite graphs. In this paper we generalize this result by obtaining the null set of complete multipartite graphs  

Let R be a commutative ring with unity. The cozero-divisor graph of R, denoted by Γ′(R), is a graph with vertex set W*(R), where W*(R) is the set of all nonzero and nonunit elements of R, and two distinct vertices a and b are adjacent if and only if a ∉ Rb and b ∉ Ra, where Rc is the ideal generated by the element c in R. Recently, it has been proved that for every nonlocal finite ring R, Γ′(R) is a unicyclic graph if and only if R ≅ ℤ2 × ℤ4, ℤ3 × ℤ3, ℤ2 × ℤ2[x]/(x 2). We generalize the aforementioned result by showing that for every commutative ring R, Γ′(R) is a unicyclic graph if and only if R ≅ ℤ2 × ℤ4, ℤ3 × ℤ3, ℤ2 × ℤ2[x]/(x 2), ℤ2[x, y]/(x, y)2, ℤ4[x]/(2x, x 2). We prove that for every... 

Let R be a commutative ring with unity. The cozero-divisor graph of R denoted by Γ'(R) is a graph with the vertex set W*(R), where W*(R) is the set of all nonzero and non-unit elements of R, and two distinct vertices a and b are adjacent if and only if a ∉ Rb and b ∉ Ra, where Rc is the ideal generated by the element c in R. Let α(Γ'(R)) and γ(Γ'(R)) denote the independence number and the domination number of Γ'(R), respectively. In this paper, we prove that if α(Γ'(R)) is finite, then R is Artinian if and only if R is Noetherian. Also, we prove that if α(Γ'(R)) is finite, then R/P is finite, for every prime ideal P. Moreover, we prove that if R is a Noetherian ring, γ(Γ'(R)) is finite and... 

Let R be a commutative ring with unity. The cozero-divisor graph of R denoted by Γ′(R) is a graph with the vertex set W*-(R), where W*-(R) is the set of all nonzero and non-unit elements of R, and two distinct vertices a and b are adjacent if and only if a*‰Rb and b*‰Ra. Let ω(Γ′(R)) and χ(Γ′(R)) denote the clique number and the chromatic number of Γ′(R), respectively. In this paper, we prove that if R is a finite commutative ring, then Γ′(R) is perfect. Also, we prove that if R is a commutative Artinian non-local ring and ω(Γ′(R)) is finite, then χ(Γ′(R)) = ω(Γ′(R)). For Artinian local ring, we obtain an upper bound for the chromatic number of cozero-divisor graph. Among other results, we... 

Let 1 < ti < t2 < ••• < th < n. A Toeplitz graph G = (V,E) denoted by Tn(tiy ..., f) is a graph where V = {1,. .. ,n} and E = {(m) I i-JI. • • >}}•this paper, we classify all regular Toeplitz graphs. Here, we present some conditions under which a Toeplitz graph has no cut-edge and cut-vertex  

The energy of a graph G, denoted by E(G), is defined as the sum of absolute values of all eigenvalues of G. A graph of order n, whose energy is less than n, i.e., E(G) < n, is said to be hypoenergetic. Graphs for which E(G) ≥ n are called non-hypoenergetic. A graph of order n is said to be orderenergetic, if its energy and its order are equal, i.e., E(G) = n. In this paper, we characterize non-hypoenergetic graphs with nullity 2. It is proved that except two graphs, every connected graph with nullity 2 is non-hypoenergetic. © 2021 University of Kragujevac, Faculty of Science. All rights reserved  

We study graphs whose adjacency matrices have determinant equal to 1 or -1, and characterize certain subclasses of these graphs. Graphs whose adjacency matrices are totally unimodular are also characterized. For bipartite graphs having a unique perfect matching, we provide a formula for the inverse of the corresponding adjacency matrix, and address the problem of when that inverse is diagonally similar to a nonnegative matrix. Special attention is paid to the case that such a graph is unicyclic. © 2006 Elsevier Inc. All rights reserved  

The energy of a graph G, denoted by E(G), is defined as the sum of absolute values of all eigenvalues of G. A graph of order n, whose energy is less than n, i.e., E(G) < n, is said to be hypoenergetic. Graphs for which E(G) ≥ n are called non-hypoenergetic. A graph of order n is said to be orderenergetic, if its energy and its order are equal, i.e., E(G) = n. In this paper, we characterize non-hypoenergetic graphs with nullity 2. It is proved that except two graphs, every connected graph with nullity 2 is non-hypoenergetic. © 2021 University of Kragujevac, Faculty of Science. All rights reserved  

Let G = (V,E) be a simple undirected graph. For a given set L ⊂ ℝ, a function ω: E → L is called an L-flow. Given a vector γ ∈ ℝv, ω is a γ-L-flow if for each υ ∈ V, the sum of the values on the edges incident to υ is γ(υ). If γ(υ) = c, for all υ ∈ V, then the γ-L-flow is called a c-sum L-flow. In this paper, the existence of γ-L-flows for various choices of sets L of real numbers is studied, with an emphasis on 1-sum flows. Let L be a subset of real numbers containing 0 and denote L*:= L {0}. Answering a question from [S. Akbari, M. Kano, and S. Zare. A generalization of 0-sum flows in graphs. Linear Algebra Appl., 438:3629-3634, 2013.], the bipartite graphs which admit a 1-sum ℝ*-flow or... 

Let G be a graph with no isolated vertex. A total dominating set of G is a set S of vertices of G such that every vertex is adjacent to at least one vertex in S. The total domatic number of a graph is the maximum number of total dominating sets which partition the vertex set of G. In this paper we provide a criterion under which a cubic graph has total domatic number at least two  

Let G be a graph with n vertices and m edges and assume that f : V ( G ) → N is a function with ∑v ∈ V ( G ) f ( v ) = m + n. We show that, if we can assign to any vertex v of G a list Lv of size f ( v ) such that G has a unique vertex coloring with these lists, then G is f-choosable. This implies that, if ∑v ∈ V ( G ) f ( v ) > m + n, then there is no list assignment L such that | Lv | = f ( v ) for any v ∈ V ( G ) and G is uniquely L-colorable. Finally, we prove that if G is a connected non-regular multigraph with a list assignment L of edges such that for each edge e = u v, | Le | = max { d ( u ), d ( v ) }, then G is not uniquely L-colorable and we conjecture that this result holds for... 

We construct counterexamples to the conjecture of Xu (1990, J. Combin. Theory Ser. B50, 319-320) that every uniquely r-colorable graph of order n with exactly (r-1)n-(r2) edges must contain a Kr. © 2001 Academic Press  

For every h ∈ ℕ, a graph G with the vertex set V (G) and the edge set E(G) is said to be h-magic if there exists a labeling l: E(G) → ℤh{0} such that the induced vertex labeling s: V (G) → ℤh, defined by s(v) = Puv∈E(G) l(uv) is a constant map. When this constant is zero, we say that G admits a zero-sum h-magic labeling. The null set of a graph G, denoted by N(G), is the set of all natural numbers h ∈ ℕ such that G admits a zero-sum h-magic labeling. In 2012, the null sets of 3-regular graphs were determined. In this paper we show that if G is an r-regular graph, then for even r (r > 2), N(G) = ℕ and for odd r (r ≠ 5), ℕ {2, 4} ⊆ N(G). Moreover, we prove that if r is odd and G is a 2-edge... 

Let R be a commutative ring with unity. The cozero-divisor graph of R denoted by Γ′(R) is a graph with the vertex set W*(R), where W*(R) is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b are adjacent if and only if a ∉ Rb and b ∉ Ra. In this paper, we show that if Γ′(R) is a forest, then Γ′(R) is a union of isolated vertices or a star. Also, we prove that if Γ′(R) is a forest with at least one edge, then R ≅ ℤ2 ⊕ F, where F is a field. Among other results, it is shown that for every commutative ring R, diam(Γ′(R[x])) = 2. We prove that if R is a field, then Γ′(R[[x]]) is totally disconnected. Also, we prove that if (R, m) is a commutative local ring and... 

A proper vertex coloring of a graph G is called a dynamic coloring if for every vertex ν with degree at least 2, the neighbors of ν receive at least two different colors. It was conjectured that if G is a regular graph, then χ2(G) - χ (G) ≤ 2. In this paper we prove that, apart from the cycles C4 and C5 and the complete bipartite graphs Kn,n, every strongly regular graph G, satisfies χ2(G) - χ (G) ≤ 1  

Let G and H be two graphs. A proper vertex coloring of G is called a dynamic coloring, if for every vertex v with degree at least 2, the neighbors of v receive at least two different colors. The smallest integer k such that G has a dynamic coloring with k colors denoted by χ2(G). We denote the cartesian product of G and H by G□H. In this paper, we prove that if G and H are two graphs and δ(G) ≥ 2, then χ2(G□H) ≤ max(χ2(G),x(H)). We show that for every two natural numbers m and n, m,n ≥ 2, χ2(Pm□Pn) = 4. Also, among other results it is shown that if 3|mn, then χ2(C m□Cn) = 3 and otherwise χ2(C m□Cn) = 4  

A Roman dominating function on a graph G = (V(G), E(G)) is a labelling f: (V(G) → {0, 1, 2} satisfying the condition that every vertex with label 0 has at least a neighbour with label 2. The Roman domination number γR(G) of G is the minimum of σv∈V(G)f(v) over all such functions. The Roman bondage number bR(G) of G is the minimum cardinality of all sets for which γR(G E) > γR(G). Recently, it was proved that for every planar graph P, bR(P) ≤ Δ(P) + 6, where Δ(P) is the maximum degree of P. We show that the Roman bondage number of every planar graph does not exceed 15 and construct infinitely many planar graphs with Roman bondage number equal to 7  

Let G be a graph and H be an abelian group. For every subset SH a map φ:E(G)→S is called an S-flow. For a given S-flow of G, and every v∈V(G), define s(v)=∑uv∈E(G)φ(uv). Let k∈H. We say that a graph G admits a k-sum S-flow if there is an S-flow such that for each vertex v,s(v)=k. We prove that if G is a connected bipartite graph with two parts X={x1,⋯,xr}, Y={y1,⋯, ys} and c1,⋯,cr,d1,⋯, ds are real numbers, then there is an R-flow such that s( xi)=ci and s(yj)=dj, for 1≤i≤r,1≤j≤s if and only if ∑i=1rci=∑j=1s dj. Also, it is shown that if G is a connected non-bipartite graph and c1,⋯,cn are arbitrary integers, then there is a Z-flow such that s(vi)=ci, for i=1, ⋯,n if and only if the number...