Dynamic Averaging Load Balancing on Cycles

We consider the following dynamic load-balancing process: given an underlying graph G with n nodes, in each step $$t\ge 0$$ t ≥ 0 , a random edge is chosen, one unit of load is created, and placed at one of the endpoints. In the same step, assuming that loads are arbitrarily divisible, the two nodes balance their loads by averaging them. We are interested in the expected gap between the minimum and maximum loads at nodes as the process progresses, and its dependence on n and on the graph structure. Peres et al. (Random Struct Algorithms 47(4):760–775, 2015) studied the variant of this process, where the unit of load is placed in the least loaded endpoint of the chosen edge, and the averaging is not performed. In the case of dynamic load balancing on the cycle of length n the only known upper bound on the expected gap is of order $$\mathcal {O}( n \log n )$$ O ( n log n ) , following from the majorization argument due to the same work. In this paper, we leverage the power of averaging and provide an improved upper bound of $$\mathcal {O} ( \sqrt{n} \log n )$$ O ( n log n ) . We introduce a new potential analysis technique, which enables us to bound the difference in load between k-hop neighbors on the cycle, for any $$k \le n/2$$ k ≤ n / 2 . We complement this with a “gap covering” argument, which bounds the maximum value of the gap by bounding its value across all possible subsets of a certain structure, and recursively bounding the gaps within each subset. We also show that our analysis can be extended to the specific instance of Harary graphs. On the other hand, we prove that the expected second moment of the gap is lower bounded by $$\Omega (n)$$ Ω ( n ) . Additionally, we provide experimental evidence that our upper bound on the gap is tight up to a logarithmic factor.

Best Graph Type to Compare Discrete Groups: Bar, Dot, and Tally

Different graph types might differ in group comparison due to differences in underlying graph schemas. Thus, this study examined whether graph schemas are based on perceptual features (i.e., each graph has a specific schema) or common invariant structures (i.e., graphs share several common schemas), and which graphic type (bar vs. dot vs. tally) is the best to compare discrete groups. Three experiments were conducted using the mixing-costs paradigm. Participants received graphs with quantities for three groups in randomized positions and were given the task of comparing two groups. The results suggested that graph schemas are based on a common invariant structure. Tally charts mixed either with bar graphs or with dot graphs showed mixing costs. Yet, bar and dot graphs showed no mixing costs when paired together. Tally charts were the more efficient format for group comparison compared to bar graphs. Moreover, processing time increased when the position difference of compared groups was increased.

The Minimum Edge Covering Energy of a Graph

In this paper, we introduce a new kind of graph energy, the minimum edge covering energy, ECE(G). It depends both on the underlying graph G, and on its particular minimum edge covering CE. Upper and lower bounds for ECE(G) are established. The minimum edge covering energy and some of the coefficients of the polynomial of well-known families of graphs like Star, Path and Cycle Graphs are computed

Biconed Graphs, Weighted Forests, and $h$-Vectors of Matroid Complexes

A well-known conjecture of Richard Stanley posits that the $h$-vector of the independence complex of a matroid is a pure ${\mathcal O}$-sequence. The conjecture has been established for various classes but is open for graphic matroids. A biconed graph is a graph with two specified 'coning vertices', such that every vertex of the graph is connected to at least one coning vertex. The class of biconed graphs includes coned graphs, Ferrers graphs, and complete multipartite graphs. We study the $h$-vectors of graphic matroids arising from biconed graphs, providing a combinatorial interpretation of their entries in terms of '$2$-weighted forests' of the underlying graph. This generalizes constructions of Kook and Lee who studied the Möbius coinvariant (the last nonzero entry of the $h$-vector) of graphic matroids of complete bipartite graphs. We show that allowing for partially $2$-weighted forests gives rise to a pure multicomplex whose face count recovers the $h$-vector, establishing Stanley's conjecture for this class of matroids. We also discuss how our constructions relate to a combinatorial strengthening of Stanley's Conjecture (due to Klee and Samper) for this class of matroids.

Beyond Rings: Gathering in 1-Interval Connected Graphs

We examine the problem of gathering [Formula: see text] agents (or multi-agent rendezvous) in dynamic graphs which may change in every round. We consider a variant of the [Formula: see text]-interval connectivity model [9] in which all instances (snapshots) are always connected spanning subgraphs of an underlying graph, not necessarily a clique. The agents are identical and not equipped with explicit communication capabilities, and are initially arbitrarily positioned on the graph. The problem is for the agents to gather at the same node, not fixed in advance. We first show that the problem becomes impossible to solve if the underlying graph has a cycle. In light of this, we study a relaxed version of this problem, called weak gathering, where the agents are allowed to gather either at the same node, or at two adjacent nodes. Our goal is to characterize the class of 1-interval connected graphs and initial configurations in which the problem is solvable, both with and without homebases. On the negative side we show that when the underlying graph contains a spanning bicyclic subgraph and satisfies an additional connectivity property, weak gathering is unsolvable, thus we concentrate mainly on unicyclic graphs. As we show, in most instances of initial agent configurations, the agents must meet on the cycle. This adds an additional difficulty to the problem, as they need to explore the graph and recognize the nodes that form the cycle. We provide a deterministic algorithm for the solvable cases of this problem that runs in [Formula: see text] number of rounds.

The Tutte polynomial and toric Nakajima quiver varieties

For a quiver $Q$ with underlying graph $\Gamma$ , we take $ {\mathcal {M}}$ an associated toric Nakajima quiver variety. In this article, we give a direct relation between a specialization of the Tutte polynomial of $\Gamma$ , the Kac polynomial of $Q$ and the Poincaré polynomial of $ {\mathcal {M}}$ . We do this by giving a cell decomposition of $ {\mathcal {M}}$ indexed by spanning trees of $\Gamma$ and ‘geometrizing’ the deletion and contraction operators on graphs. These relations have been previously established in Hausel–Sturmfels [6] and Crawley-Boevey–Van den Bergh [3], however the methods here are more hands-on.

Locating Eigenvalues of a Symmetric Matrix whose Graph is Unicyclic

We present a linear-time algorithm that computes in a given real interval the number of eigenvalues of any symmetric matrix whose underlying graph is unicyclic. The algorithm can be applied to vertex- and/or edge-weighted or unweighted unicyclic graphs. We apply the algorithm to obtain some general results on the spectrum of a generalized sun graph for certain matrix representations which include the Laplacian, normalized Laplacian and signless Laplacian matrices.

The minimum vertex–vertex dominating Laplacian energy of a graph

Let [Formula: see text] denote the set of all blocks of a graph [Formula: see text]. Two vertices are said to vv-dominate each other if they are vertices of the same block. A set [Formula: see text] is said to be vertex–vertex dominating set (vv-dominating set) if every vertex in [Formula: see text] is vv-dominated by some vertex in [Formula: see text]. The vv-domination number [Formula: see text] is the cardinality of the minimum vv-dominating set of [Formula: see text]. In this paper, we introduce new kind of graph energy, the minimum vv-dominating Laplacian energy of a graph denoting it as LE[Formula: see text]. It depends both on the underlying graph of [Formula: see text] and the particular minimum vv-dominating set of [Formula: see text]. Upper and lower bounds for LE[Formula: see text] are established and we also obtain the minimum vv-dominating Laplacian energy of some family of graphs.

Unmixed and Cohen–Macaulay Weighted Oriented Kőnig Graphs

Let D be a weighted oriented graph, whose underlying graph is G, and let I (D) be its edge ideal. If G has no 3-, 5-, or 7-cycles, or G is Kőnig, we characterize when I (D) is unmixed. If G has no 3- or 5-cycles, or G is Kőnig, we characterize when I (D) is Cohen–Macaulay. We prove that I (D) is unmixed if and only if I (D) is Cohen–Macaulay when G has girth greater than 7 or G is Kőnig and has no 4-cycles.