In this paper, we introduce a new method to compute magnitude homology of general graphs. To each direct sum component chain complexes, assign pair simplicial complexes whose complex is isomorphic it. First states our main theorem specialized trees, which gives another proof for the known fact that trees are diagonal. After that, consider graphs, may have cycles. We also demonstrate some comput...

We introduce a unifying approach for invariants of finite matroids that count mappings to set. The aim this paper is show if the cardinalities with fixed values on restricted set satisfy contraction–deletion rules, then there relation among them can be expressed in terms linear algebra. In way, we study regular chain groups, nowhere-zero flows and tensions graphs, acyclic totally cyclic orienta...

let $n$ be any positive integer, the friendship graph $f_n$ consists of $n$ edge-disjoint triangles that all of them meeting in one vertex. a graph $g$ is called cospectral with a graph $h$ if their adjacency matrices have the same eigenvalues. recently in href{http://arxiv.org/pdf/1310.6529v1.pdf}{http://arxiv.org/pdf/1310.6529v1.pdf} it is proved that if $g$ is any graph cospectral with $f_n$...

Generalizing results of Temperley [11], Brooks, Smith, Stone and Tutte [1] and others [10, 7] we describe a natural equivalence between three planar objects: weighted bipartite planar graphs; planar Markov chains; and tilings with convex polygons. This equivalence provides a measure-preserving bijection between dimer coverings of a weighted bipartite planar graph and spanning trees of the corre...

Real world graphs have been observed to display a number of surprising properties. These properties include heavy-tails for inand out-degree distributions, small diameters, and a densification law [5]. These features do not arise from the classical Erdos-Renyi random graph model [1]. To address these difficulties, Kronecker Graphs were first introduced in [5] as a new method of generating graph...

For each graph and each positive integer n, we define a chain complex whose graded Euler characteristic is equal to an appropriate nspecialization of the dichromatic polynomial. This also gives a categorification of n-specializations of the Tutte polynomial of graphs. Also, for each graph and integer n ≤ 2, we define the different one variable n-specializations of the dichromatic polynomials, a...

Probabilistic graphical models are today one of the most well used architec-tures for modelling and reasoning about knowledge with uncertainty. Themost widely used subclass of these models is Bayesian networks that hasfound a wide range of applications both in industry and research. Bayesiannetworks do however have a major limitation which is that only asymmetricrelationship...

We consider uniform random walks on finite graphs with n nodes. When the hitting times are symmetric, the expected covering time is at least 89 log n O(n log log n) uniformly over all such graphs. We also obtain bounds for the covering times in terms of the eigenvalues of the transition matrix of the Markov chain. For distance-regular graphs, a general lower bound of ( n 1 ) l o g n is obtained...

We investigate the eigenvalue spacing distributions for randomly generated 4-regular Cayley graphs on SL2(Fp) by numerically calculating their spectra. We present strong evidence that the distributions are Poisson and hence do not follow the Gaussian orthogonal ensemble. Among the Cayley graphs of SL2(Fp) we consider are the new expander graphs recently discovered by Y. Shalom. In addition, we ...

Given a semigraph, we can construct graphs Sa, Sca, Se and S1e. In the same pattern, we construct bipartite graphs CA(S), A(S), VE(S), CA(S) and A(S). We find the equality of domination parameters in the bipartite graphs constructed with the domination and total domination parameters of the graphs Sa and Sca. We introduce the domination and independence parameters for the bipartite semigraph. W...