Threshold graphs are generated from one node by repeatedly adding a node that links to all existing nodes or adding a node without links. In the weighted threshold graph, we add a new node in step i, which is linked to all existing nodes by a link of weight w_i . In this work, we consider the set A_N that contains all Laplacian matrices of weighted threshold graphs of order N. We show that A_N forms a commutative algebra. Using this, we find a common basis of eigenvectors for the matrices in A_N . It follows that the eigenvalues of each matrix in A_N can be represented as a linear transformation of the link weights. In addition, we prove that, if there are just three or fewer different weights, two weighted threshold graphs with the same Laplacian spectrum must be isomorphic.

Degree-based graph construction is a fundamental problem in network science. A graph is simple if there are no self-loops and no multiple links between any pair of nodes in the graph. A degree sequence d = (d1 , d2, ... , dN) is graphical if d can be represented as the degree sequence of at least one simple graph, where the graph is called a realization of the sequence d. In this work, we introduce a novel method (LSFGR) for generating simple graphs from graphical degree sequences, focusing additionally on connectedness and on assortativity. LSFGR guarantees connected graphs for all potentially connected degree sequences. In the case where a degree sequence has no simple realization, LSFGR produces graphs with at most one node with self-loops. In addition, the graphs generated from LSFGR characterize real-world networks with medium assortativity.