in {xx, yy} if and only if xy is an edge in E. Such a word w is called G’s **k**-**11**-representant. A uniform (resp., t-uniform) representation of a graph G is a word, satisfying the required properties, in which each letter occurs the same (resp., t) number of times. As is stated above, in this paper we assume V to be [n] = {1, 2, . . . , n} for some n ≥ 1. Note that 0-**11**- **representable** **graphs** are precisely word-**representable** **graphs**, and that 0-**11**-representants are precisely word-representants. We also note that the “**11**” in “**k**-**11**-**representable**” refers to counting occurrences of the consecutive pattern **11** in the word induced by a pair of letters {x, y}, which is exactly the total number of occurrences of the factors in {xx, yy}. Throughout the paper, we normally omit the word “consecutive” in “consecutive pattern” for brevity. Finally, we let G (**k**) denote the class of **k**-**11**-**representable** **graphs**.

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we raise some concerns about Conjecture 7, while confirming it for **graphs** on at most 9 vertices. In Section 3 we present a complementary computational approach using constraint programming, enabling us count connected non-word-**representable** **graphs**. In particular, in Section 3 we report that using 3 years of CPU time, we found out that 64.65% of all connected **graphs** on **11** vertices are non-word-**representable**. Another important corollary of our results in Section 3 is the correction of the published result [19, 20] on the number of connected non- word-**representable** **graphs** on 9 vertices (see Table 2). In Section 4 we introduce the notion of a **k**-semi-transitive orientation refining the notion of a semi-transitive orientation, and show that 3-semi-transitively orientable **graphs** are not necessarily semi-transitively orientable. Finally, in Section 5 we suggest a few directions for further research and experimentation.

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we raise some concerns about Conjecture 7, while confirming it for **graphs** on at most 9 vertices. In Section 3 we present a complementary computational approach using constraint programming, enabling us count connected non-word-**representable** **graphs**. In particular, in Section 3 we report that using 3 years of CPU time, we found out that 64.65% of all connected **graphs** on **11** vertices are non-word-**representable**. Another important corollary of our results in Section 3 is the correction of the published result [19, 20] on the number of connected non- word-**representable** **graphs** on 9 vertices (see Table 2). In Section 4 we introduce the notion of a **k**-semi-transitive orientation refining the notion of a semi-transitive orientation, and show that 3-semi-transitively orientable **graphs** are not necessarily semi-transitively orientable. Finally, in Section 5 we suggest a few directions for further research and experimentation.

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In [2], Ahmad et al. determined the exact value of super edge magic defi- ciency of (n, t)-kite graph for all odd n; t ≡ 0, 1 (mod 4) and also showed the upper bound for all odd n, t ≡ 2, 3 (mod 4). In [4], Ahmad et al. determined the upper bound for all odd n and t ≡ 3, 7 (mod 8), t 6= **11**. In the next lemma, we determined the upper bound for t = **11**.

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, Then Rosa [10] called the mapping the β-labeling (valuation) of a graph G, Golomb [4] subsequently called such labeling to be graceful labeling and the graph is called a graceful graph, while is called an induced edge’s graceful labeling. -graceful labeling is the generalization of graceful labeling that introduced by Slater [**11**] in 1982 and by Maheo and Thuillier [8] in 1982.

Lemma 2.2. Let S be a subset of vertices of G=Cir(n,B) with **k** ≥ 3 and G[S] has no isolated vertices. If |S| is odd, then S dominates at most (2k+3) |S| - (**k**+1) vertices of G. Proof: Let S a be subset of vertices of G with |S|= t, where t is odd. Without loss of generality we may assume that G[S] has d = ( |:|!"

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It seems that the approximation becomes better for in- creasing M , but the data are not sufficient for a reliable extrapolation. A formal repetition of our earlier argu- ment regarding the limit M → ∞ fails because trans- lationally invariant systems will still retain five relevant hopping parameters that determine the density. On the other hand, particle conservation only helps to reduce the number of effectively independent density coefficients to six, so it seems probable that even in this limit the density is not noninteracting v **representable**. We have performed calculations for closed, translationally invari- ant ring chains up to M = 9 that are governed by the same ratio of significant hopping parameters and den- sity coefficients. None of these systems was found to be noninteracting v **representable**.

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So far, there have been many results on the verica- tion of Conjecture 1.1 in the literature, especially for **graphs** with particular structures. In [6,10,11], the lin- ear **k**− arboricity of trees are studied. In [12,13,14], the linear **k**− arboricity and the linear arboricity of some regular **graphs** are studied. In [15,16,17,18], the linear 2-arboricity of planar **graphs** are obtained and the linear **k**− arboricity of cubic **graphs** are ob- tained. In [2,3,4,9,19], the linear **k**− arboricity of the balanced complete multipartite **graphs** **K** n(m) , **K** n,n , **K** n ,

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In [7], Kulli introduced the first and second **K** Hyper Banhatti indices and it was defined as ( ) ∑ , ( ) ( )- . (3) ( ) ∑ , ( ) ( )- . (4) Here, we introduce some new Topological indices based on eccentricity as follows.

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Reuse See Attached Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the[r]

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Step 2: Let us assume that the result is true for **k** = t. Then the result is proved for **k** = t + 1. Consider the graph BF(5(t+1)). From Recursive Construction 2 [8], BF(5t+5) is decomposed into 2 5 copies of BF(5t) without wings and the last 5 levels form a pattern of BF(5) with 2 5 vertex groups where each vertex group has 2 5t vertices.

suppose the line graph L(G) is **k**-minimally nonouterplanar. Then L(G) is planar. By Theorem C, ∆ (G) ≤4 and every point of degree 4 is a cutpoint. Since G is a block, so it has no cutpoint. Thus if ∆ (G) =4, then L(G) is nonplanar, a contradiction. Thus ∆ (G) ≤3. As the graph G is a m-minimally nonouterplanr (p, q) block, by Lemma 1, it has (q − p+m) nonboundary lines in the plane embedding of G. Again by Lemma 2, these (q − p+m) nonboundary lines of G correspond to the inner points of L(G). It implies that L(G) is (q − p+m)-minimally nonouterplanar. Thus the condition q − p+m=**k** holds.

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“ Graceful labeling “was introduced by Rosa (1967)[7]. J.A. Gallian [3] studied a complete survey on graph labeling . David .W and Anthony. E. Baraaukas [1] have investigate the cycle structure of Fibonacci graceful **graphs**. A Fibonacci graceful labeling and Super Fibonacci graceful labeling have been introduced by Kathiresan and Amutha [5] in 2006. N. Murugesan and R. Uma [6] have obtained some Cycle- related **graphs** under Fibonacci graceful.

Stallmann et al. [SBG01] in testing of various heuristics proposed the idea of testing them on a wide range of random and isomorphism classes. A random class of **graphs** consists of some number of **graphs** (usually 32, 64 or 128), generated in such a way that all instances in the class have the same number of nodes and edges and the **graphs** are sometimes similar in other ways as well. An isomorphism class consists of **graphs** which are random presentations of the same graph G. The name isomorphism derives from the fact that every graph in an isomorphism class is isomorphic to every other graph. The graph differ only in (a) the order of appearance of the edges in the input file, and (b) the initial order of nodes on each layer as specified by an auxiliary ordering file.

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[1] introduced notion of cordial labeling of **graphs**. The cocept of **k**-difference cordial graph was introduced in [4] . Recently Ponraj etal [5] has been introduced the concept of **k**-total prime cordial graph . Motivated by this, we introduce **k**-total difference cordial labeling of **graphs**. Also we prove that every graph is a subgraph of a connected **k**-total differnce cordial **graphs** and investigate 3-total prime cordial labeling of sevarel **graphs** like path,star, bistar, complete bipartite graph etc .

In 1965, Lofti A.Zadeh[3] introduced the concept of a fuzzy subset of a set as a method for representing the phenomena of uncertainty in real life situation. Azriel Rosenfeld introduced fuzzy **graphs** in 1975[3], which is growing fast and has numerous applications in various fields. Nagoor Gani and Radha [2] introduced regular fuzzy **graphs**, total degree and totally regular fuzzy **graphs**. Alison Northup [1] studied some properties on (2, **k**)- regular **graphs** in her bachelor thesis. N.R Santhi Maheswari and C. Sekar introduced d 2 of a vertex in **graphs** [4] and (3, **k**)-

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By a graph, we mean a finite, undirected graph without loops and multiple edges, for terms not defined here, we refer to Harary [4]. For standard terminology and notations related to number theory we refer to Burton [2] and graph labeling, we refer to Gallian [3]. The notion of prime labeling for **graphs** originated with Roger Entringer and was introduced in a paper by Tout et al. [8] in the early 1980s and since then it is an active field of research for many scholars. Patel et al.[6] introduce the notion of neighborhood-prime labeling of graph and they present the neighborhood-prime labeling of various **graphs** in [6,7]. Ananthavalli et al. present the neighborhood-prime labeling of some special **graphs** in [1]. In [9], Vaidya et al. introduce the concept of **k**-prime labeling of **graphs**. Lawrence et al. introduce the notation of **k**-neighborhood- prime labeling and they present the neighborhood-prime labeling of G ∗ B B, where B is the book with triangular and rectangle pages, G

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O((log d) √ d) around u ∗ . We first run the chains for a burn-in period of T = O(n) steps. By Lemma 4.1 with high probability (in d) the disagreements are contained in this local ball B around u ∗ . Hence we can focus attention inside this local ball B (with high probability). Since the volume of this ball is not too large, by Theorem 4.1 all of the low degree vertices have the local uniformity property and they maintain it for O(n) steps. Hence for **k** > αd we get contraction for disagreements at low degree vertices. Since the vertices at the boundaries of the block are all low degree vertices and these are the vertices with non-zero weight dist() in our path coupling analysis as in the proof of Theorem 3.1 for the **k** > 2∆ case, then we get that the expected distance dist() contracts in every step. Since the number of disagreements is not too large (by the second part of Lemma 4.1) after O(n) steps we get that the expected weight is small, and we can conclude that the mixing time is O(N log N ).

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We consider a variant of the standard edge cover problem. Let G be a graph with vertex set and edge set . We use [6] for terminology and notation which are not defined here and consider only simple **graphs** without isolated vertices. For every nonempty subset of , the subgraph of G whose vertex set is the set of vertices of the edges in

i and iii: the equivalence of i and iii can be deduced from the max-flow min-cut theorem 10, **11**. Convert G X, Y ; E into a network by a adding a source vertex s with **k** multiedges between s and each vertex x ∈ X; b adding a sink vertex t with **k** multiedges between t and each vertex y ∈ Y ; and c orienting each edge into a directed arc going from s to X, from X to Y , or from Y to t see Figure 2. Clearly, G has a **k**-factor ⇔ the network has a kn-flow from s to t ⇔ any cut in the network that separates s and t contains at least kn forward edges. For any S ⊆ X and T ⊆ Y , consider the cut shown in dashed line in Figure 2, we have

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