Abstract. In this paper the idea of sum distance which is a metric, in a fuzzy graph is introduced. The concepts of eccentricity, radius, diameter, center and self centered fuzzy graphs are studied using this metric. Some properties of eccentric nodes, peripheral nodes and central nodes are obtained. A characterization of self centered complete fuzzy graph is obtained and conditions under which...

For a simple graph G, let e(G) denote the number of edges and Sk(G) denote the sum of the k largest eigenvalues of the signless Laplacian matrix of G. We conjecture that for any graph G with n vertices, Sk(G) ≤ e(G) + k+1 2 for k = 1, . . . , n. We prove the conjecture for k = 2 for any graph, and for all k for regular graphs. The conjecture is an analogous to a conjecture by A.E. Brouwer with ...

Orbital acceleration in polar: a = (r̈ − rθ̇2)r̂ + (rθ̈ + 2ṙθ̇)θ̂ Legendre polynomials: P1(x) = 1, P2(x) = x, P3(x) = (3x 2 − 1)/2; orthogonality: ∫ 1 −1 Pm(x)Pn(x)dx = 2 2n+1δmn Eccentricity: e > 1 hyperbola; e = 1 parabola; 0 < e < 1 ellipse; e = 0 circle Stirling’s Approximation: lnN ! = N lnN −N Gaussian Integral: ∫∞ −∞ e −ax2dx = √ π/a ∫∞ −∞ x 2e−ax 2 dx = √ π/2a3/2 ∫∞ 0 x 3e−αx 2 = 1/2α2 Delta ...

The pseudo-determinant Det(A) of a square matrix A is defined as the product of the nonzero eigenvalues of A. It is a basis-independent number which is up to a sign the first nonzero entry of the characteristic polynomial of A. We prove Det(FG) = ∑ P det(FP)det(GP) for any two n×m matrices F,G. The sum to the right runs over all k × k minors of A, where k is determined by F and G. If F = G is t...

Abstract. We prove that the inverse of a positive-definite matrix can be approximated by a weighted-sum of a small number of matrix exponentials. Combining this with a previous result [6], we establish an equivalence between matrix inversion and exponentiation up to polylogarithmic factors. In particular, this connection justifies the use of Laplacian solvers for designing fast semi-definite pr...

the wiener index w(g) of a connected graph g is defined as the sum of the distances betweenall unordered pairs of vertices of g. the eccentricity of a vertex v in g is the distance to avertex farthest from v. in this paper we obtain the wiener index of a graph in terms ofeccentricities. further we extend these results to the self-centered graphs.

&lt;abstract&gt;&lt;p&gt;Let $ A(G) and D(G) be the adjacency matrix degree diagonal of a graph G $, respectively. For any real number \alpha \in[0, 1] Nikiforov defined A_{\alpha} $-matrix as A_{\alpha}(G) = D(G)+(1-\alpha)A(G) $. Let S_k(A_{\alpha}(G)) sum k largest eigenvalues In this paper, some bounds on are obtained, which not only extends results signless Laplacian matrix, but it also gi...

Abstract Let G be a simple connected graph of order n and size m. The matrix L(G)= D(G)− A(G) is called the Laplacian G,where D(G) are degree diagonal adjacency matrix, respectively. vertex sequence d1 ≥ d2 ≥··· dn let μ1 μ2 μ n−1 &gt;μn = 0 eigenvalues G. invariants, energy (LE), Laplacian-energy-like invariant (LEL) Kirchhoff index (Kf), defined in terms G, as <m:math xmlns:m="http://www.w3.o...

Given a connected graph G, the Randić index R(G) is the sum of 1 √ d(u)d(v) over all edges {u, v} of G, where d(u) and d(v) are the degree of vertices u and v respectively. Let q(G) be the largest eigenvalue of the singless Laplacian matrix of G and n = |V (G)|. Hansen and Lucas (2010) made the following conjecture:

The m-connectivity index χα(G) of an organic molecule whose molecular graph is G is the sum of the weights (di1di2 ...dim+1) , where i1−i2− ...−im+1 runs over all paths of length m in G and di denotes the degree of vertex vi. We find upper bounds for χα(G) when m ≥ 1 and α ≥ −1 (α 6= 0) using the eigenvalues of the Laplacian matrix of an associated weighted graph.