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• ##### The Line n Sigraph of a Symmetric n Sigraph IV

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An n-tuple (a^sub 1^, a^sub 2^, ... , a^sub n^) is symmetric, if a^sub n-k+1^, 1 ≤ k ≤ n. Let H^sub n^ = (a^sub 1^, a^sub 2^, ... , a^sub n^) : a^sub k^ ∈ +, - }, a^sub k^ = a^sub n-k+1^, 1 ≤ k ≤ n} be the set of all symmetric n-tuples. Asymmetric n-sigraph (symmetric n-marked graph) is an ordered pair S^sub n^ = (G, σ) (S^sub n^ = (G, μ)), where G = (V, E) is a graph called the underlying graph of S^sub n^ and σ : E [arrow right] H^sub n^ (μ : V [arrow right] H^sub n^) is a function. In Bagga et al. (1995) introduced the concept of the super line graph of index r of a graph G, denoted by L^sub r^(G). The vertices of L^sub r^(G) are the r-subsets of E(G) and two vertices P and Q are adjacent if there exist p ∈ P and q ∈ Q such that p and q are adjacent edges in G. Analogously, one can define the super line symmetric n-sigraph of index r of a symmetric n-sigraph S^sub n^ = (G, σ) as a symmetric n-sigraph L^sub r^(S^sub n^) = (L^sub r^(G), σ'), where L^sub r^(G) is the underlying graph of L^sub r^(S^sub n^), where for any edge PQ in L^sub r^(S^sub n^), σ'(PQ) = σ(P)σ(Q). It is shown that for any symmetric n-sigraph S^sub n^, its L^sub r^(S^sub n^) is i-balanced and we offer a structural characterization of super line symmetric n-sigraphs of index r. Further, we characterize symmetric n-sigraphs S^sub n^ for which S^sub n^ ~ L^sub 2^(S^sub n^) ~ L^sub 2^(S^sub n^) ~ L(S^sub n^) and L^sub 2^(S^sub n^) ~ ... where ~ denotes switching equivalence and L^sub 2^(S^sub n^), L(S^sub n^) and ... are denotes the super line symmetric n-sigraph of index 2, line symmetric n-sigraph and complementary symmetric n-sigraph of S^sub n^ respectively. Also, we characterize symmetric n-sigraphs S^sub n^ for which S^sub n^ [congruent with] L^sub 2^(S^sub n^) and L^sub 2^(S^sub n^) [congruent with] L(S^sub n^). [PUBLICATION ABSTRACT]
• ##### 3-Product Cordial Labeling of Some Graphs

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A mapping f : V(G) [arrow right] {0, 1, 2} is called a 3-product cordial labeling if |v^sub f^(i) - v^sub f^(j)| ≤ 1 and |e^sub f^(i) - e^sub f^(j)| ≤ 1 for any i, j ∈ {0, 1, 2}, where v^sub f^(i) denotes the number of vertices labeled with i, e^sub f^(i) denotes the number of edges xy with f(x)f(y) ≡ i (mod 3). A graph with a 3-product cordial labeling is called a 3-product cordial graph. In this paper, we establish that the duplicating arbitrary vertex in cycle C^sub n^, duplicating arbitrarily edge in cycle C^sub n^, duplicating arbitrary vertex in wheel W^sub n^, Ladder L^sub n^, Triangular Ladder TL^sub n^ and the graph [left angle bracket]W^sub n^^sup (1)^] : W^sub n^^sup (2)^) : ... : W^sub n^^sup (k)^[right angle bracket] are 3-product cordial. [PUBLICATION ABSTRACT]
• ##### Domination in Transformation Graph G^sup +-+^

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Let G = (V, E) be a simple undirected graph of order n and size m. The transformation graph of G is a simple graph with vertex set V(G) ∪ E(G) in which adjacency is defined as follows: (a) two elements in V(G) are adjacent if and only if they are adjacent in G (b) two elements in E(G) are adjacent if and only if they are non-adjacent in G and (c) one element in V(G) and one element in E(G) are adjacent if and only if they are incident in G. It is denoted by G^sup +-+^. A set S ⊆ V(G) is a dominating set if every vertex in V - S is adjacent to at least one vertex in S. The minimum cardinality taken over all dominating sets of G is called the domination number of G and is denoted by γ(G). In this paper, we investigate the domination number of transformation graph. We determine the exact values for some standard graphs and obtain several bounds. Also we prove that for any connected graph G of order n ≥ 5, γ(G^sup +-+^) ≤ ... . [PUBLICATION ABSTRACT]
• ##### Equations for Spacelike Biharmonic General Helices with Timelike Normal According to Bishop Frame in The Lorentzian Group of Rigid Motions E(1, 1)

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In this paper, we study spacelike biharmonic general helices according to Bishop frame in the Lorentzian group of rigid motions E(1, 1). We characterize the spacelike biharmonic general helices in terms of their curvatures in the Lorentzian group of rigid motions E(1, 1). [PUBLICATION ABSTRACT]
• ##### Super Mean Labeling of Some Classes of Graphs

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Let G be a (p, q) graph and f : V(G) [arrow right] {1, 2, 3, . . . , p + q} be an injection. For each edge e = uv, let f*(e) = (f(u) + f(v))/2 if f(u) + f(v) is even and f*(e) = (f(u) + f(v) + 1)/2 if f(u) + f(v) is odd. Then f is called a super mean labeling if f(V) ∪ {f* (e) : e ∈ E(G)} = 1, 2, 3, . . . , p+q}. A graph that admits a super mean labeling is called a super mean graph. In this paper we prove that ... generalized antiprism A^sup m^^sub n^ and the double triangular snake D(T^sub n^) are super mean graphs. [PUBLICATION ABSTRACT]
• ##### Laplacian Energy of Certain Graphs

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Let G be a graph with n vertices and m edges. Let μ^sub 1^, μ^sub 2^, ..., μ^sub n^ be the eigenvalues of the Laplacian matrix of G. The Laplacian energy .... In this paper, we calculate the exact Laplacian energy of complete graph, complete bipartite graph, path, cycle and friendship graph. [PUBLICATION ABSTRACT]
• ##### The t Pebbling Number of Jahangir Graph

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Given a configuration of pebbles on the vertices of a connected graph G, a pebbling move (or pebbling step) is defined as the removal of two pebbles from a vertex and placing one pebble on an adjacent vertex. The t-pebbling number, f^sub t^(G) of a graph G is the least number m such that, however m pebbles are placed on the vertices of G, we can move t pebbles to any vertex by a sequence of pebbling moves. In this paper, we determine f^sup t^(G) for Jahangir graph J^sub 2,m^. [PUBLICATION ABSTRACT]
• ##### Matrix Representation of Biharmonic Curves in Terms of Exponential Maps in the Special Three-Dimensional [straight phi]-Ricci Symmetric Para-Sasakian Manifold

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In this paper, we study biharmonic curves in the special three-dimensional [straight phi]-Ricci symmetric para-Sasakian manifold P. We construct matrix representation of biharmonic curves in terms of exponential maps in the special three-dimensional [straight phi]-Ricci symmetric para-Sasakian manifold P. [PUBLICATION ABSTRACT]
• ##### Bounds on Szeged and PI Indexes in terms of Second Zagreb Index

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In this short note, we studied the vertex version and the edge version of the Szeged index and the PI index and obtained bounds for these indices in terms of the Second Zagreb index. Also, established the connections of bounds to the above sighted indices. [PUBLICATION ABSTRACT]
• ##### On Finsler Spaces with Unified Main Scalar LC of the Form L sup 2 C sup 2 f y g x

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In the year 1979, M. Matsumoto was studying Finsler spaces with vanishing T-tensor and come to know that for such Finsler spaces L^sup 2^C^sub 2^ is a function of x only. Later on in the year 1980, Matsumoto with Numata concluded that the condition L^sup 2^C^sup 2^ = f(x) is not sufficient for vanishing of T-tensor. F. Ikeda in the year 1984, studied Finsler spaces whose L^sup 2^C^sup 2^ is function of x in detail. In the present paper we shall discuss a Finsler space for which L^sup 2^C^sup 2^ is a function of x and y (y^sup i^ = x^sup i^) in the form L^sup 2^C^sup 2^ = f(y) + g(x). [PUBLICATION ABSTRACT]
• ##### On Square Difference Graphs

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In graph theory number labeling problems play vital role. Let G = (V, E) be a (p, q)-graph with vertex set V and edge set E. Let f be a vertex valued bijective function from V(G) [arrow right] {0, 1,..., p - 1}. An edge valued function f* can be defined on G as a function of squares of vertex values. Graphs which satisfy the injectivity of this type of edge valued functions are called square graphs. Square graphs have two major divisions: they are square sum graphs and square difference graphs. In this paper we concentrate on square difference or SD graphs. An edge labeling f* on E(G) can be defined as follows. f*(uv) = |(f(u))^sup 2^ - (f(v))^sup 2^| for every uv in E(G). If f* is injective, then the labeling is said to be a SD labeling. A graph which satisfies SD labeling is known as a SD graph. We illuminate some of the results on number theory into the structure of SD graphs. Also, established some classes of SD graphs and established that every graph can be embedded into a SD graph. [PUBLICATION ABSTRACT]
• ##### On Fuzzy Matroids

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The aim of this paper is to discuss properties of fuzzy regular-flats, fuzzy C-flats, fuzzy alternative-sets and fuzzy i-flats. Moreover, we characterize some peculiar fuzzy matroids via these notions. Finally, we provide a decomposition of fuzzy strong maps. [PUBLICATION ABSTRACT]
• ##### NOTES FROM EXECUTIVE FOR ENLISTED MATTERS

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To bridge this gap we will look at possibly sending 7-levels back to Silver Flag for refresher training or delivering disbursing training via Defense Connect Online sessions where instructors can train, answer questions, and test individuals. [...]thank you for your service and what you do to take care of our Airmen every day.
• ##### DIRECTOR, DEFENSE FINANCIAL MANAGEMENT AND COMPTROLLER SCHOOL

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Recent Congressional budget cuts and workforce reductions have forced our schools to more thoroughly examine how we deliver courses to our customers without degrading the quality of the education and training. [...]in our FY13 schedule, we added three DDSC classes.
• ##### RECOGNITION: ACES HIGH AWARDS

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While in technical training at Keesler AFB he served as a white rope, the backbone of the community center for all tech school students at Keesler AFB. In 2006, SSgt Duchesne was hand selected to fill a selectively manned position at the United States Air Force Academy Financial Management Directorate.
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