A k-digraph is an orientation of a multi-graph that is without loops and contains at most k edges between any pair of distinct vertices. So 1-digraph is an oriented graph, and complete 1-digraph is a tournament. We obtain necessary and sufficient conditions for a sequence of non-negative integers in non-decreasing order to be a sequence of numbers, called marks, attached to vertices of 2-digraph.
For a finite commutative ring Z n with identity 1 = 0, the zero divisor graph Γ(Z n ) is a simple connected graph having vertex set as the set of non-zero zero divisors, where two vertices x and y are adjacent if and only if xy = 0. We find the normalized Laplacian spectrum of the zero divisor graphs Γ(Z n ) for various values of n and characterize n for which Γ(Z n ) is normalized Laplacian integral. We also obtain bounds for the sum of graph invariant S * β (G)-the sum of the β-th power of the non-zero normalized Laplacian eigenvalues of Γ(Z n ).
Let G be a simple graph with n vertices, m edges and having adjacency eigenvalues λ 1 , λ 2 ,. .. , λ n. The energy E(G) of the graph G is defined as E(G) = n i=1 |λ i |. In this paper, we obtain the upper bounds for the energy E(G) in terms of the vertex covering number τ , the clique number ω, the number of edges m, maximum vertex degree d 1 and second maximum vertex degree d 2 of the connected graph G. These upper bounds improve some of the recently known upper bounds.
Let A = (a1, a2, ..., an) be a degree sequence of a simple bipartite graph. We present an algorithm that takes A as input, and outputs a simple bipartite realization of A, without stalling. The running time of the algorithm is ⊝(n1n2), where ni is the number of vertices in the part i of the bipartite graph. Then we couple the generation algorithm with a rejection sampling scheme to generate a simple realization of A uniformly at random. The best algorithm we know is the implicit one due to Bayati, Kim and Saberi (2010) that has a running time of O(mamax), where
$m = {1 \over 2}\sum\nolimits_{i = 1}^n {{a_i}}
and amax is the maximum of the degrees, but does not sample uniformly. Similarly, the algorithm presented by Chen et al. (2005) does not sample uniformly, but nearly uniformly. The realization of A output by our algorithm may be a start point for the edge-swapping Markov Chains pioneered by Brualdi (1980) and Kannan et al.(1999).
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