15 research outputs found
Strongly distance-balanced graphs and graph products
AbstractA graph G is strongly distance-balanced if for every edge uv of G and every i≥0 the number of vertices x with d(x,u)=d(x,v)−1=i equals the number of vertices y with d(y,v)=d(y,u)−1=i. It is proved that the strong product of graphs is strongly distance-balanced if and only if both factors are strongly distance-balanced. It is also proved that connected components of the direct product of two bipartite graphs are strongly distance-balanced if and only if both factors are strongly distance-balanced. Additionally, a new characterization of distance-balanced graphs and an algorithm of time complexity O(mn) for their recognition, where m is the number of edges and n the number of vertices of the graph in question, are given
2-Edge Distance-Balanced Graphs
In a graph A, for each two arbitrary vertices g, h with
d(g,h)=2,|MAg2h|=mAg2h is introduced the number of edges of A that are closer
to g than to h. We say A is a 2-edge distance-balanced graph if we have
mAg2h=mAh2g. In this article, we verify the concept of these graphs and present
a method to recognize k-edge distance-balanced graphs for k = 2,3 using
existence of either even or odd cycles. Moreover, we investigate situations
under which the Cartesian and lexicographic products lead to 2-edge distance
-balanced graphs. In some subdivision-related graphs 2-edge distance-balanced
property is verified
A new lower bound for doubly metric dimension and related extremal differences
In this paper a new graph invariant based on the minimal hitting set problem
is introduced. It is shown that it represents a tight lower bound for the
doubly metric dimension of a graph. Exact values of new invariant for paths,
stars, complete graphs and complete bipartite graph are obtained. The paper
analyzes some tight bounds for the new invariant in general case. Also several
extremal differences between some related invariants are determined
New transmission irregular chemical graphs
The transmission of a vertex of a (chemical) graph is the sum of
distances from to other vertices in . If any two vertices of have
different transmissions, then is a transmission irregular graph. It is
shown that for any odd number there exists a transmission irregular
chemical tree of order . A construction is provided which generates new
transmission irregular (chemical) trees. Two additional families of chemical
graphs are characterized by property of transmission irregularity and two
sufficient condition provided which guarantee that the transmission
irregularity is preserved upon adding a new edge
On distance-balanced generalized Petersen graphs
A connected graph of diameter is
-distance-balanced if for every with
, where is the set of vertices of that are closer
to than to . We prove that the generalized Petersen graph is
-distance-balanced provided that is large enough
relative to . This partially solves a conjecture posed by Miklavi\v{c} and
\v{S}parl \cite{Miklavic:2018}. We also determine when
is large enough relative to
Consensus strategies for signed profiles on graphs
The median problem is a classical problem in Location Theory: one searches for a
location that minimizes the average distance to the sites of the clients. This is for desired
facilities as a distribution center for a set of warehouses. More recently, for obnoxious
facilities, the antimedian was studied. Here one maximizes the average distance to the
clients. In this paper the mixed case is studied. Clients are represented by a profile, which
is a sequence of vertices with repetitions allowed. In a signed profile each element is
provided with a sign from (+,-). Thus one can take into account whether the client
prefers the facility (with a + sign) or rejects it (with a - sign). The graphs for which all
median sets, or all antimedian sets, are connected are characterized. Various consensus
strategies for signed profiles are studied, amongst which Majority, Plurality and Scarcity.
Hypercubes are the only graphs on which Majority produces the median set for all signed
profiles. Finally, the antimedian sets are found by the Scarcity Strategy on e.g. Hamming
graphs, Johnson graphs and halfcubes