Let G be a simple graph with vertex set V (G).. Let nucleus(G) and diadem(G) be the intersection and union, respectively, of all maximum size critical independent sets in G. In this paper, we will give two new characterizations of König-Egerváry graphs involving nucleus(G) and diadem(G). We also prove a related lower bound for the independence number of a graph. This work answers several conjectures posed by Jarden, Levit, and Mandrescu.
Joint degree vectors give the number of edges between vertices of degree i and degree j for 1 ≤ i ≤ j ≤ n − 1 in an n-vertex graph. We find lower and upper bounds for the maximum number of nonzero elements in a joint degree vector as a function of n. This provides an upper bound on the number of estimable parameters in the exponential random graph model with bidegree-distribution as its sufficient statistics.
A matching M of a graph G is maximal if it is not a proper subset of any other matching in G. Maximal matchings are much less known and researched than their maximum and perfect counterparts. In particular, almost nothing is known about their enumerative properties. In this paper we present the recurrences and generating functions for the sequences enumerating maximal matchings in two classes of chemically interesting linear polymers: polyspiro chains and benzenoid chains. We also analyze the asymptotic behavior of those sequences and determine the extremal cases.
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