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In this paper, we show that every (3k−3)-edge-connected graph G, under a certain degree condition, can be edge-decomposed into k factors G1,…,Gk such that for each vertex v∈V(Gi), |dGi(v)−dG(v)/k|<1, where 1≤i≤k. As an application, we deduce that every 6-edge-connected graph G can be edge-decomposed into three factors G1, G2, and G3 such that for each vertex v∈V(Gi) with 1≤i≤3, |dGi(v)−dG(v)/3|<1, unless G has exactly one vertex z with dG(z)⁄≡30. Next, we show that every odd-(3k−2)-edge-connected graph G can be edge-decomposed into k factors G1,…,Gk such that for each vertex v∈V(Gi), dGi(v) and dG(v) have the same parity and |dGi(v)−dG(v)/k|<2, where k is an odd positive integer and 1≤i≤k.... 

An induced path factor of a graph (Formula presented.) is a set of induced paths in (Formula presented.) with the property that every vertex of (Formula presented.) is in exactly one of the paths. The induced path number (Formula presented.) of (Formula presented.) is the minimum number of paths in an induced path factor of (Formula presented.). We show that if (Formula presented.) is a connected cubic graph on (Formula presented.) vertices, then (Formula presented.). Fix an integer (Formula presented.). For each (Formula presented.), define (Formula presented.) to be the maximum value of (Formula presented.) over all connected (Formula presented.) -regular graphs (Formula presented.) on... 

A signed graph Gσ is an ordered pair (V(G),E(G)), where V(G) and E(G) are the set of vertices and edges of G, respectively, along with a map σ that signs every edge of G with +1 or −1. An eigenvalue of the associated adjacency matrix of Gσ, denoted by A(Gσ), is a main eigenvalue if the corresponding eigenspace has a non-orthogonal eigenvector to the all-one vector j. We conjectured that for every graph G≠K2,K4{e}, there is a switching σ such that all eigenvalues of Gσ are main. We show that this conjecture holds for every Cayley graphs, distance-regular graphs, vertex and edge-transitive graphs as well as double stars and paths. © 2020 Elsevier Inc  

A Deza graph G with parameters (n,k,b,a) is a k-regular graph with n vertices such that any two distinct vertices have b or a common neighbours. The children GA and GB of a Deza graph G are defined on the vertex set of G such that every two distinct vertices are adjacent in GA or GB if and only if they have a or b common neighbours, respectively. A strongly Deza graph is a Deza graph with strongly regular children. In this paper we give a spectral characterisation of strongly Deza graphs, show relationships between eigenvalues, and study strongly Deza graphs which are distance-regular. © 2021 Elsevier B.V  

An induced path factor of a graph (Formula presented.) is a set of induced paths in (Formula presented.) with the property that every vertex of (Formula presented.) is in exactly one of the paths. The induced path number (Formula presented.) of (Formula presented.) is the minimum number of paths in an induced path factor of (Formula presented.). We show that if (Formula presented.) is a connected cubic graph on (Formula presented.) vertices, then (Formula presented.). Fix an integer (Formula presented.). For each (Formula presented.), define (Formula presented.) to be the maximum value of (Formula presented.) over all connected (Formula presented.) -regular graphs (Formula presented.) on... 

A signed graph Gσ is an ordered pair (V(G),E(G)), where V(G) and E(G) are the set of vertices and edges of G, respectively, along with a map σ that signs every edge of G with +1 or −1. An eigenvalue of the associated adjacency matrix of Gσ, denoted by A(Gσ), is a main eigenvalue if the corresponding eigenspace has a non-orthogonal eigenvector to the all-one vector j. We conjectured that for every graph G≠K2,K4{e}, there is a switching σ such that all eigenvalues of Gσ are main. We show that this conjecture holds for every Cayley graphs, distance-regular graphs, vertex and edge-transitive graphs as well as double stars and paths. © 2020 Elsevier Inc  

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... 

An induced path factor of a graph (Formula presented.) is a set of induced paths in (Formula presented.) with the property that every vertex of (Formula presented.) is in exactly one of the paths. The induced path number (Formula presented.) of (Formula presented.) is the minimum number of paths in an induced path factor of (Formula presented.). We show that if (Formula presented.) is a connected cubic graph on (Formula presented.) vertices, then (Formula presented.). Fix an integer (Formula presented.). For each (Formula presented.), define (Formula presented.) to be the maximum value of (Formula presented.) over all connected (Formula presented.) -regular graphs (Formula presented.) on... 

A signed graph Gσ is an ordered pair (V(G),E(G)), where V(G) and E(G) are the set of vertices and edges of G, respectively, along with a map σ that signs every edge of G with +1 or −1. An eigenvalue of the associated adjacency matrix of Gσ, denoted by A(Gσ), is a main eigenvalue if the corresponding eigenspace has a non-orthogonal eigenvector to the all-one vector j. We conjectured that for every graph G≠K2,K4{e}, there is a switching σ such that all eigenvalues of Gσ are main. We show that this conjecture holds for every Cayley graphs, distance-regular graphs, vertex and edge-transitive graphs as well as double stars and paths. © 2020 Elsevier Inc  

The energy of a graph G, E(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. Gutman et al. proved that for every cubic graph of order n, E(G)≥n. Here, we improve this result by showing that if G is a connected subcubic graph of order n≥8, then E(G)≥1.01n. Also, we prove that if G is a traceable subcubic graph of order n≥8, then E(G)>1.1n. Let G be a connected cubic graph of order n≥8, it is shown that E(G)>n+2. It was proved that if G is a connected cubic graph of order n, then E(G)≤1.65n. Also, in this paper we would like to present the best lower bound for the energy of a connected cubic graph. We introduce an infinite family of connected cubic graphs whose for... 

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  

Let G be a 2k-edge-connected graph with 𝑘 ≥ 0 and let 𝐿(𝑣) ⊆ {𝑘,…, 𝑑𝐺(𝑣)} for every 𝑣 ∈ 𝑉 (𝐺). A spanning subgraph F of G is called an L-factor, if 𝑑𝐹 (𝑣) ∈ 𝐿(𝑣) for every 𝑣 ∈ 𝑉 (𝐺). In this article, we show that if (Formula presented.) for every 𝑣 ∈ 𝑉 (𝐺), then G has a k-edge-connected L-factor. We also show that if 𝑘 ≥ 1 and (Formula presented.) for every 𝑣 ∈ 𝑉 (𝐺), then G has a k-edge-connected L-factor. © 2018 Wiley Periodicals, Inc  

A graph G is called uniquelyk-list colorable (UkLC) if there exists a list of colors on its vertices, say L= { Sv∣ v∈ V(G) } , each of size k, such that there is a unique proper list coloring of G from this list of colors. A graph G is said to have propertyM(k) if it is not uniquely k-list colorable. Mahmoodian and Mahdian (Ars Comb 51:295–305, 1999) characterized all graphs with property M(2). For k≥ 3 property M(k) has been studied only for multipartite graphs. Here we find bounds on M(k) for graphs embedded on surfaces, and obtain new results on planar graphs. We begin a general study of bounds on M(k) for regular graphs, as well as for graphs with varying list sizes. © 2018, Springer... 

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  

A universal labeling of a graph G is a labeling of the edge set in G such that in every orientation (Formula presented.) of G for every two adjacent vertices v and u, the sum of incoming edges of v and u in the oriented graph are different from each other. The universal labeling number of a graph G is the minimum number k such that G has universal labeling from (Formula presented.) denoted it by (Formula presented.). We have (Formula presented.), where (Formula presented.) denotes the maximum degree of G. In this work, we offer a provocative question that is: “Is there any polynomial function f such that for every graph G, (Formula presented.)?”. Towards this question, we introduce some... 

Let G be a graph. Assume that l and k are two natural numbers. An l-sum flow on a graph G is an assignment of non-zero real numbers to the edges of G such that for every vertex v of G the sum of values of all edges incident with v equals l. An l-sum k-flow is an l-sum flow with values from the set {±1,…, ±(k — 1)}. Recently, it was proved that for every r, r ≥ 3, r ≠ 5, every r-regular graph admits a 0-sum 5-flow. In this paper we settle a conjecture by showing that every 5-regular graph admits a 0-sum 5-flow. Moreover, we prove that every r-regular graph of even order admits a 1-sum 5-flow  

Liu and Xu (1998) and Ellingham, Nam and Voss (2002) independently showed that every k-edge-connected simple graph G has a spanning tree T such that for each vertex v, dT(v) ≤ ⌈ d(v)/k ⌉ + 2. In this paper we show that every k-edge-connected graph G has a spanning tree T such that for each vertex v, dT(v)≤ ⌈ d(v)-2/k ⌉ + 2; also if G has k edge-disjoint spanning trees, then T can be found such that for each vertex v, dT(v) ≤ ⌈ d(v)-1/k ⌉ + 1. This result implies that every (r-1)-edge-connected r-regular graph (with r ≥ 4) has a spanning Eulerian subgraph whose degrees lie in the set {2,4,6}; also reduces the edge-connectivity needed for some theorems due to Barát and Gerbner (2014) and... 

Let R be a commutative ring, let Z(R) be the set of all zero-divisors of R and Reg(R) = RZ(R). The regular graph of R, denoted by G(R), is a graph with all elements of Reg(R) as the vertices, and two distinct vertices x, y ∈ Reg(R) are adjacent if and only if x+y ∈ Z(R). In this paper we show that if R is a commutative Noetherian ring and 2 ∈ Z(R), then the chromatic number and the clique number of G(R) are the same and they are 2n, where n is the minimum number of prime ideals whose union is Z(R). Also, we prove that all trees that can occur as the regular graph of a ring have at most two vertices  

A 2-hued coloring of a graph G is a coloring such that, for every vertex v∈V(G) of degree at least 2, the neighbors of v receive at least two colors. The smallest integer k such that G has a 2-hued coloring with k colors is called the 2-hued chromatic number of G, and is denoted by χ2(G). In this paper, we will show that, if G is a regular graph, then χ2(G)-χ(G)≤2log 2(α(G))+3, and, if G is a graph and δ(G)<2, then χ2(G)-χ(G)≤1+4 Δ2δ-1⌉(1+log 2Δ(G)2Δ(G)-δ(G)(α(G))), and in the general case, if G is a graph, then χ2(G)-χ(G)≤2+min α′(G),α(G)+ω(G)2  

The impact of adding correlation to a population of neurons on the information and the activity of the population is one of the fundamental questions in recent system neuroscience. In this paper, we would like to introduce topology-based correlation at the level of storing patterns in a recurrent network. We then study the effects of topological patterns on the activity and memory capacity of the network. The general aim of the present work is to show how the repertoire of possible stored patterns is determined by the underlying network topology.Two topological probability rules for pattern selection in recurrent network are introduced. The first one selects patterns according to a...