This work introduces a highly scalable spectral graph densification (SGL) framework for learning resistor networks with linear measurements, such as node voltages and currents. We show that the proposed approach is equivalent to solving classical graphical Lasso problems Laplacian-like precision matrices. prove given <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="ht...

A linear forest is a graph that connected components are chordless paths. A linear partition of a graph G is a partition of its edge set into linear forests and la(G) is the minimum number of linear forests in a linear partition. It is well known that la(G) = 2 when G is a cubic graph and Wormald [17] conjectured that if |V (G)| ≡ 0 (mod 4), then it is always possible to find a linear partition...

In the extension of irregularity indices, Abdo et. al. [1] defined the total irregu-larity of a graph G = (V, E) as irrt(G) = 21 Pu,v∈V (G) du − dv, where du denotesthe vertex degree of a vertex u ∈ V (G). In this paper, we investigate the totalirregularity of trees with bounded maximal degree Δ and state integer linear pro-gramming problem which gives standard information about extremal trees a...

A linear graph is a graph whose vertices are totally ordered. Biological and linguistic sequences with interactions among symbols are naturally represented as linear graphs. Examples include protein contact maps, RNA secondary structures and predicate-argument structures. Our algorithm, linear graph miner (LGM), leverages the vertex order for efficient enumeration of frequent subgraphs. Based o...

We consider the complexity of the emptiness problem for various classes of graph languages deened by eNCE (edge label neighborhood controlled embedding) graph grammars. In particular, we show that the emptiness problem is undecidable for general eNCE graph grammars, DEXPTIME-complete for connuent and boundary eNCE graph grammars, PSPACE-complete for linear eNCE graph grammars, NL-complete for d...

Many real-life networks can be modelled by stochastic processes with a spatial embedding. In such processes, the link probability decreases with distance. Using the theory of graph limits, we show how to recognize graph sequences produced by random graph processes with a linear embedding (a natural embedding into R). We define an operator Γ which applies to graph limits, which assumes the value...

Mader asked whether every C4-free graph G contains a subdivision of a complete graph whose order is at least linear in the average degree of G. We show that there is a subdivision of a complete graph whose order is almost linear. More generally, we prove that every Ks,t-free graph of average degree r contains a subdivision of a complete graph of order r 1 2 + 1 2(s−1).