for any $k in mathbb{n}$, the $k$-subdivision of graph $g$ is a simple graph $g^{frac{1}{k}}$, which is constructed by replacing each edge of $g$ with a path of length $k$. in [moharram n. iradmusa, on colorings of graph fractional powers, discrete math., (310) 2010, no. 10-11, 1551-1556] the $m$th power of the $n$-subdivision of $g$ has been introduced as a fractional power of $g$, denoted by ...

We extend the Lp theory of sparse graph limits, which was introduced in a companion paper, by analyzing different notions of convergence. Under suitable restrictions on node weights, we prove the equivalence of metric convergence, quotient convergence, microcanonical ground state energy convergence, microcanonical free energy convergence, and large deviation convergence. Our theorems extend the...

In power law graphs, cut quality varies inversely with cut balance. Using some million node social graphs as a test bed, we empirically investigate this property and its implications for graph partitioning. We use six algorithms, including Metis and MQI (state of the art methods for finding bisections and quotient cuts) and four relaxation/rounding methods. We find that an SDP relaxation avoids...

For a connected graph G, we derive tight inequalities relating the smallest signless Laplacian eigenvalue to the largest normalised Laplacian eigenvalue. We investigate how vectors yielding small values of the Rayleigh quotient for the signless Laplacian matrix can be used to identify bipartite subgraphs. Our results are applied to some graphs with degree sequences approximately following a pow...

Let Γ be a finite G-symmetric graph whose vertex set admits a non-trivial Ginvariant partition B with block size v. A framework for studying such graphs Γ was developed by Gardiner and Praeger which involved an analysis of the quotient graph ΓB relative to B, the bipartite subgraph Γ[B,C] of Γ induced by adjacent blocks B,C of ΓB and a certain 1-design D(B) induced by a block B ∈ B. The present...

For a simple connected graph G of order n, the normalized Laplacian is square matrix defined as [Formula: see text], where text] diagonal whose i-th entry text]. In this paper, we find eigenvalues joined union regular graphs in terms adjacency and quotient associated with G. finite group power which two distinct vertices are by an edge if only one other. As consequence graphs, investigate cyclic

In the present paper, we study the continuous-time quantum walk on quotient graphs. On such graphs, there is a straightforward reduction of problem to a subspace that can be considerably smaller than the original one. Along the lines of reductions, by using the idea of calculation of the probability amplitudes for continuous-time quantum walk in terms of the spectral distribution associated wit...

The permanent-determinant method and its generalization, the HafnianPfaffian method, are methods to enumerate perfect matchings of plane graphs that were discovered by P. W. Kasteleyn. We present several new techniques and arguments related to the permanent-determinant with consequences in enumerative combinatorics. Here are some of the results that follow from these techniques: 1. If a biparti...

We study how the number c(X) of components of a graph X can be expressed through the number and properties of the components of a quotient graph X/∼ . We partially rely on classic qualifications of graph homomorphisms such as locally constrained homomorphisms and on the concept of equitable partition and orbit partition. We introduce the new definitions of pseudo-covering homomorphism and of co...

The k-th power of a n-vertex graph X is the iterated cartesian product of X with itself. The k-th symmetric power of X is the quotient graph of certain subgraph of its k-th power by the natural action of the symmetric group. It is natural to ask if the spectrum of the k-th power –or the spectrum of the k-th symmetric power– is a complete graph invariant for small values of k, for example, for k...