- 601 University Drive
Department of Mathematics
Texas State University
San Marcos, TX 78666
U.S.A. - (+1) 512-245-3743 (Phone)
Daniela Ferrero
Texas State University, Mathematics, Faculty Member
Research Interests:
For a given graph G = (V,E) and a positive integer k, the super line graph of index k of G is the graph Sk(G) which has for vertices all the k-subsets of E(G), and two vertices S and T are adjacent whenever there exist s in S and t in T... more
For a given graph G = (V,E) and a positive integer k,
the super line graph of index k of G is the graph Sk(G)
which has for vertices all the k-subsets of E(G), and two
vertices S and T are adjacent whenever there exist s in S
and t in T such that s and t share a common vertex. In
the super line multigraph Lk(G) we have an adjacency for
each such occurrence.
We give a formula to find the adjacency matrix of Lk(G).
If G is a regular graph, we calculate all the eigenvalues of
Lk(G) and their multiplicities. From those results we give
an upper bound on the number of isolated vertices.
the super line graph of index k of G is the graph Sk(G)
which has for vertices all the k-subsets of E(G), and two
vertices S and T are adjacent whenever there exist s in S
and t in T such that s and t share a common vertex. In
the super line multigraph Lk(G) we have an adjacency for
each such occurrence.
We give a formula to find the adjacency matrix of Lk(G).
If G is a regular graph, we calculate all the eigenvalues of
Lk(G) and their multiplicities. From those results we give
an upper bound on the number of isolated vertices.
Research Interests:
For a given digraph G = (V,E) and a positive integer k, the super line digraph of index k of G is the digraph Sk(G) which has for vertices all the k-subsets of E(G), and two vertices S and T are adjacent whenever there exist edges in the... more
For a given digraph G = (V,E) and a positive integer k, the super line digraph of index k of G is the digraph Sk(G) which has for vertices all the k-subsets of E(G), and two vertices S and T are adjacent whenever there exist edges in the form
(u,v) in S and (v,w) in T for some u, v, w in V .
The super line digraph is a generalization of the
super line graph. Indeed, if the digraph G is symmetric, the super line digraph of G is isomorphic to the super line graph of the graph obtained by removing the orientation of the edges of G. We study the link between properties of super line digraphs and super line graphs.
(u,v) in S and (v,w) in T for some u, v, w in V .
The super line digraph is a generalization of the
super line graph. Indeed, if the digraph G is symmetric, the super line digraph of G is isomorphic to the super line graph of the graph obtained by removing the orientation of the edges of G. We study the link between properties of super line digraphs and super line graphs.
Research Interests:
Intuitively, a signed graph is a graph in which every edge is labeled with a + or ¡ sign. For each edge, its sign represents the mode of the relationship between the vertices it joins. In a signed graph, cycles can be naturally given the... more
Intuitively, a signed graph is a graph in which every edge is labeled with a + or ¡ sign. For each edge, its sign represents the mode of the relationship between the vertices it joins.
In a signed graph, cycles can be naturally given the sign corresponding to the product of its edges.
Then, a signed graph is called balanced when all
the cycles have positive sign. Balanced signed graph have multiple applications in the field of social networks. Consequently, there is a significant amount of research in the problem of determining if a signed graph is balanced or not. In particular, some authors investigated extensions to signed graph of the line graph and studied under what circumstances the signed graphs obtained are balanced or not.
This paper presents a new operation, which is
also an extension to signed graphs of the line
graph, with the property that applied to any signed
graph always produces a balanced signed graph.
In a signed graph, cycles can be naturally given the sign corresponding to the product of its edges.
Then, a signed graph is called balanced when all
the cycles have positive sign. Balanced signed graph have multiple applications in the field of social networks. Consequently, there is a significant amount of research in the problem of determining if a signed graph is balanced or not. In particular, some authors investigated extensions to signed graph of the line graph and studied under what circumstances the signed graphs obtained are balanced or not.
This paper presents a new operation, which is
also an extension to signed graphs of the line
graph, with the property that applied to any signed
graph always produces a balanced signed graph.
Research Interests:
For a graph G, the P2-path graph, P2(G), has for vertices the set of all paths of length 2 in G. Two vertices are connected when their union is a path or a cycle of length 3. We present lower bounds on the edge-connectivity of the P2 path... more
For a graph G, the P2-path graph, P2(G), has for vertices the set of all paths of length 2 in
G. Two vertices are connected when their union is a path or a cycle of length 3. We present
lower bounds on the edge-connectivity of the P2 path graph of a connected graph G and give conditions
for maximum connectivity. A maximally edge-connected graph is super-lambda if each minimum edge
cut is trivial, and it is optimum super-lambda if each minimum nontrivial edge cut consists of all the
edges adjacent to one edge. We give conditions on G, for P2(G) to be super-lambda and optimum
super-lambda.
G. Two vertices are connected when their union is a path or a cycle of length 3. We present
lower bounds on the edge-connectivity of the P2 path graph of a connected graph G and give conditions
for maximum connectivity. A maximally edge-connected graph is super-lambda if each minimum edge
cut is trivial, and it is optimum super-lambda if each minimum nontrivial edge cut consists of all the
edges adjacent to one edge. We give conditions on G, for P2(G) to be super-lambda and optimum
super-lambda.
Research Interests:
An interconnection network is a highly symmetrical connected graph of order n nodes, size m edges, connectivity k and diameter d, where n and k are large but m and d are small. Many interconnection networks are defined algebraically in... more
An interconnection network is a highly symmetrical connected graph of order n nodes, size m edges, connectivity k
and diameter d, where n and k are large but m and d are small. Many interconnection networks are defined
algebraically in such a way that each node has an integer value. Then every edge can be assigned the sum of the
two nodes it joins. These numbers are called the edge sums of the graph. The edge sum problem of a graph is to
characterize the set of edge sums. This problem was introduced by Graham and Harary who presented the
solution for hypercubes. Our object is to characterize the edge sums for another family of interconnection
networks, namely, deBruijn graphs.
and diameter d, where n and k are large but m and d are small. Many interconnection networks are defined
algebraically in such a way that each node has an integer value. Then every edge can be assigned the sum of the
two nodes it joins. These numbers are called the edge sums of the graph. The edge sum problem of a graph is to
characterize the set of edge sums. This problem was introduced by Graham and Harary who presented the
solution for hypercubes. Our object is to characterize the edge sums for another family of interconnection
networks, namely, deBruijn graphs.
Research Interests:
Research Interests:
This work contains the most important results of the doctoral thesis "Graphs and Hypergraphs as Interconnection Network Models" by Daniela Ferrero supervised by Carles Padro. This thesis was done at the Department of Applied Mathematics... more
This work contains the most important results of the doctoral thesis "Graphs and Hypergraphs as Interconnection Network Models" by Daniela Ferrero supervised by Carles Padro. This thesis was done at the Department of Applied Mathematics and Telematics of the PolytechnicUniversity of Catalonia (Spain).
We can divide this work into three parts: In the first one some results about the fault-tolerance of known models for point-to-point networks based on digraphs are
given. The second part is devoted to the study of hyperdigraphs based models of bus networks. This is a new area so before some equivalent results about the fault-tolerance, we need to prove
other topological properties. Finally, a little part concerning with random
sequences useful in stream cipher applications is presented.
We can divide this work into three parts: In the first one some results about the fault-tolerance of known models for point-to-point networks based on digraphs are
given. The second part is devoted to the study of hyperdigraphs based models of bus networks. This is a new area so before some equivalent results about the fault-tolerance, we need to prove
other topological properties. Finally, a little part concerning with random
sequences useful in stream cipher applications is presented.
Research Interests:
A generalized p-cycle is a digraph whose set of vertices is partitioned in p parts that can be ordered in such a way that a vertex is adjacent only to vertices in the next part. The families of BGC(p,d,dk) and KGC(p, d, dp+k+dk ) are the... more
A generalized p-cycle is a digraph whose set of vertices
is partitioned in p parts that can be ordered in
such a way that a vertex is adjacent only to vertices
in the next part. The families of BGC(p,d,dk) and
KGC(p, d, dp+k+dk ) are the largest known p-cyles for
their degree and diameter.
In this paper we present routing algorithms for both
families. Such algorithms route over paths of length at
most the value of the diameter plus two units. Moreover,
this bound is attained only in the case that the
number of faulty elements (nodes or links) is maxim,um.
is partitioned in p parts that can be ordered in
such a way that a vertex is adjacent only to vertices
in the next part. The families of BGC(p,d,dk) and
KGC(p, d, dp+k+dk ) are the largest known p-cyles for
their degree and diameter.
In this paper we present routing algorithms for both
families. Such algorithms route over paths of length at
most the value of the diameter plus two units. Moreover,
this bound is attained only in the case that the
number of faulty elements (nodes or links) is maxim,um.
Research Interests:
The partial line digraph technique was introduced in [7] in order to construct digraphs with a minimum diameter, maximum connectivity, and good expandability. To find a new method to construct directed hypergraphs with a minimum diameter,... more
The partial line digraph technique was introduced in
[7] in order to construct digraphs with a minimum diameter,
maximum connectivity, and good expandability.
To find a new method to construct directed hypergraphs
with a minimum diameter, we present in this paper an adaptation of that technique to directed hypergraphs. Directed hypergraphs are used as models for interconnection networks whose vertices are
linked by directed buses. The connectivity and expandability
of partial line directed hypergraphs are studied.
Besides, we prove a conjecture by J-C. Bermond
and F. Ergincan about the characterization of line
directed hypergraphs.
[7] in order to construct digraphs with a minimum diameter,
maximum connectivity, and good expandability.
To find a new method to construct directed hypergraphs
with a minimum diameter, we present in this paper an adaptation of that technique to directed hypergraphs. Directed hypergraphs are used as models for interconnection networks whose vertices are
linked by directed buses. The connectivity and expandability
of partial line directed hypergraphs are studied.
Besides, we prove a conjecture by J-C. Bermond
and F. Ergincan about the characterization of line
directed hypergraphs.
Research Interests:
The aim of this paper is to lower bound the connectivity of k-path graphs. From the bounds obtained, we give conditions to guarantee maximum connectivity. Then, it is shown that those maximally connected graphs satisfying the previous... more
The aim of this paper is to lower bound the connectivity of k-path graphs. From the bounds obtained, we give conditions to guarantee maximum connectivity. Then, it is shown that those maximally connected graphs satisfying the previous conditions are also super-\lambda. While doing so, we derive some properties about the girth and the diameter of path graphs. Finally, the results are extended to path graphs resulting from the iteration of the k-path graph operator.
Research Interests:
Electric power networks must be continuously monitored. Such monitoring can be eciently accomplished by placing phase measurement units (PMUs) at selected network locations. Due to the high cost of the PMUs, their number must be... more
Electric power networks must be continuously monitored. Such monitoring
can be eciently accomplished by placing phase measurement units (PMUs)
at selected network locations. Due to the high cost of the PMUs, their number
must be minimized. The power domination problem consists of nding
the minimum number of PMUs needed to monitor a given electric power
system. The power dominating problem is NP-hard, but closed formulas
for the power domination number of certain networks, such as rectangular
meshes [4] have been found. In this work we extend the results for rectangular
meshes to honeycomb meshes.
can be eciently accomplished by placing phase measurement units (PMUs)
at selected network locations. Due to the high cost of the PMUs, their number
must be minimized. The power domination problem consists of nding
the minimum number of PMUs needed to monitor a given electric power
system. The power dominating problem is NP-hard, but closed formulas
for the power domination number of certain networks, such as rectangular
meshes [4] have been found. In this work we extend the results for rectangular
meshes to honeycomb meshes.
Research Interests:
Let G be a simple non-complete graph of order n. The r-component edge connectivity of G denoted as λr(G) is the minimum number of edges that must be removed from G in order to obtain a graph with (at least) r connected components. The... more
Let G be a simple non-complete graph of order n. The r-component edge connectivity of G denoted as λr(G) is the minimum number of edges that must be removed from G in order to obtain a graph with (at least) r connected components. The concept of r-component edge connectivity generalizes that of edge connectivity by taking into account the number of components of the resulting graph. In this paper we establish bounds of the r component edge connectivity of an important family of interconnection network models, the generalized Petersen graphs GP(n, k) in which n and k are relatively prime integers.
Research Interests:
The P3 intersection graph of a graph G has for vertices all the induced paths of order 3 in G. Two vertices in P3(G) are adjacent if the corresponding paths in G are not disjoint. A w-container between two different vertices u and v in... more
The P3 intersection graph of a graph G has for vertices all the induced paths
of order 3 in G. Two vertices in P3(G) are adjacent if the corresponding paths in G are not
disjoint.
A w-container between two different vertices u and v in a graph G is a set of w internally
vertex disjoint paths between u and v. The length of a container is the length of the
longest path in it. The w-wide diameter of G is the minimum number l such that there is
a w-container of length at most l between any pair of different vertices u and v in G.
Interconnection networks are usually modeled by graphs. The w-wide diameter provides
a measure of the maximum communication delay between any two nodes when up to w −1
nodes fail. Therefore, the wide diameter constitutes a measure of network fault tolerance.
In this paper we construct containers in P3(G) and apply the results obtained to the
study of their connectivity and wide diameters.
of order 3 in G. Two vertices in P3(G) are adjacent if the corresponding paths in G are not
disjoint.
A w-container between two different vertices u and v in a graph G is a set of w internally
vertex disjoint paths between u and v. The length of a container is the length of the
longest path in it. The w-wide diameter of G is the minimum number l such that there is
a w-container of length at most l between any pair of different vertices u and v in G.
Interconnection networks are usually modeled by graphs. The w-wide diameter provides
a measure of the maximum communication delay between any two nodes when up to w −1
nodes fail. Therefore, the wide diameter constitutes a measure of network fault tolerance.
In this paper we construct containers in P3(G) and apply the results obtained to the
study of their connectivity and wide diameters.
Let G be a digraph, L(G) its line digraph and A(G) and A(LG) their adjacency matrices. We present relations between the Jordan Normal Form of these two matrices. In addition, we study the spectra of those matrices and obtain a... more
Let G be a digraph, L(G) its line digraph and A(G) and A(LG) their adjacency matrices. We present relations between the Jordan Normal Form of these two matrices. In addition, we study the spectra of those matrices and obtain a relationship between their characteristic polynomials that allows us to relate properties of G and L(G), specifically the number of cycles of a given length.
Let G1 and G2 be copies of a graph G, and let f be a function from V (G1) to V (G2). Then a functigraph C(G, f) = (V,E) is a generalization of a permutation graph, where V = V (G1) U V (G2) and E = E(G1) U E(G2) U {uv : u in V (G1), v... more
Let G1 and G2 be copies of a graph G, and let f be a function from V (G1) to V (G2). Then a functigraph C(G, f) = (V,E) is a generalization of a permutation graph, where V = V (G1) U V (G2) and E = E(G1) U E(G2) U {uv : u in V (G1), v in (G2),
v = f(u)}. In this paper, we study colorability and planarity of functigraphs.
v = f(u)}. In this paper, we study colorability and planarity of functigraphs.
For an arbitrary simple graph G and a positive integer r, the super line multigraph of index r of G, denoted Mr(G), has for vertices all the r-subsets of edges. Two vertices S and T are joined by as many edges as pairs of distinct edges s... more
For an arbitrary simple graph G and a positive integer r, the super line multigraph of index r of G, denoted Mr(G), has for vertices all the r-subsets of edges. Two vertices S and T are joined by as many edges as pairs of distinct edges s in S and t in T share a common vertex in G. We present spectral properties of Mr(G) and particularly, if G is a regular graph, we calculate all the eigenvalues of Mr(G) and their multiplicities in terms of those of G.
Research Interests:
Directed hypergraphs are used to model networks whose nodes are connected by directed buses. We study in this paper two parameters related to the fault-tolerance of directed bus networks: the connectivity and the fault-diameter of... more
Directed hypergraphs are used to model networks whose nodes are connected by directed buses. We study in this paper two parameters related to the fault-tolerance of directed bus networks: the connectivity and the fault-diameter of directed hypergraphs. Some bounds are given for those parameters. As a consequence, we obtain that de Bruijn-Kautz directed hypergraphs and, more generally, iterated line directed hypergraphs provide models for highly fault-tolerant directed bus networks.
Research Interests:
A set S of vertices is defined to be a power dominating set (PDS) of a graph G if every vertex and every edge in G can be monitored by the set S according to a set of rules for power system monitoring. The minimum cardinality of a PDS of... more
A set S of vertices is defined to be a power dominating set (PDS) of a graph G if every vertex and every edge in G can be monitored by the set S according to a set of rules for power system monitoring. The minimum cardinality of a PDS of G is its power domination number. In this article, we find upper bounds for the power domination number of some families of Cartesian products of graphs: the cylinders PnxCm for integers n ≥ 2, m ≥ 3, and the tori CnxCm for integers n,m ≥ 3.We apply similar techniques to present upper bounds for the power domination
number of generalized Petersen graphs P(m, k ). We prove those upper bounds provide the exact values of the power domination numbers if the integersm, n, and k satisfy some given relations.
number of generalized Petersen graphs P(m, k ). We prove those upper bounds provide the exact values of the power domination numbers if the integersm, n, and k satisfy some given relations.
Simple modeling with polyominoes raises intriguing
questions about evolution.
questions about evolution.
Research Interests:
The diameter of a graph is an important factor for communication as it determines the maximum communication delay between any pair of processors in a network. Graham and Harary [N. Graham, F. Harary, Changing and unchanging the diameter... more
The diameter of a graph is an important factor for communication as it determines the maximum communication delay between any pair of processors in a network. Graham and Harary [N. Graham, F. Harary, Changing and unchanging the diameter of a hypercube,
Discrete Applied Mathematics 37/38 (1992) 265–274] studied how the diameter of hypercubes can be affected by increasing and decreasing edges. They concerned whether the diameter is changed or remains unchanged when the edges are increased or decreased. In this paper, we modify three measures proposed in Graham and Harary (1992) to include the extent of the change of the diameter. Let D^{-k}(G) be the least number of edges whose addition to G decreases the diameter by (at least) k, D^{+0}(G) is the maximum number of edges whose deletion from G does not change the diameter, and D^{+k}(G) is the least number of edges whose deletion from G increases the diameter by (at least) k. In this paper, we find the values of D^{+k}(C_m), D^{-1}(T_{m;n}), D^{-2}(T_{m;n}), D^{+1}(T_{m;n}), and a lower bound for D^{+0}(T_{m;n}) where C_m is a cycle with m vertices, and T_{m;n} is a torus of size m by n.
Discrete Applied Mathematics 37/38 (1992) 265–274] studied how the diameter of hypercubes can be affected by increasing and decreasing edges. They concerned whether the diameter is changed or remains unchanged when the edges are increased or decreased. In this paper, we modify three measures proposed in Graham and Harary (1992) to include the extent of the change of the diameter. Let D^{-k}(G) be the least number of edges whose addition to G decreases the diameter by (at least) k, D^{+0}(G) is the maximum number of edges whose deletion from G does not change the diameter, and D^{+k}(G) is the least number of edges whose deletion from G increases the diameter by (at least) k. In this paper, we find the values of D^{+k}(C_m), D^{-1}(T_{m;n}), D^{-2}(T_{m;n}), D^{+1}(T_{m;n}), and a lower bound for D^{+0}(T_{m;n}) where C_m is a cycle with m vertices, and T_{m;n} is a torus of size m by n.
Iterated line digraphs arise naturally in designing fault tolerant systems. Diameter vulnerability measures the increase in diameter of a digraph when some of its vertices or arcs fail. Thus, the study of diameter vulnerability is a... more
Iterated line digraphs arise naturally in designing fault tolerant systems. Diameter vulnerability measures the increase in diameter of a digraph when some of its vertices or arcs fail. Thus, the study of diameter vulnerability is a suitable approach to the fault tolerance of a network. In this article we present some upper bounds for diameter vulnerability of iterated line digraphs L^k(G). Our bounds depend basically on the girth of the digraph G and on the number of iterations k. These bounds generalize some previous results on diameter vulnerability of line digraphs. Also, we apply our results to several important families of line digraphs such as Kautz digraphs and deBruijn generalized cycles, which contain deBruijn digraphs, the Reddy-Pradhan-Kuhl digraphs, and the butterflies. Our bounds allow us to obtain improvements in known results on diameter vulnerability for all these families.
A generalized p-cycle is a digraph whose set of vertices is partitioned in p parts that can be ordered in such a way that a vertex is adjacent only to vertices in the next part. The families of $BGC(p,d,d^p)$ and $KGC(p,d,d^{p+k}+d^p)$... more
A generalized p-cycle is a digraph whose set of vertices is partitioned in p parts that can be ordered in such a way that a vertex is adjacent only to vertices in the next part. The families of $BGC(p,d,d^p)$ and $KGC(p,d,d^{p+k}+d^p)$ are the largest known p-cycles for their degree and diameter. In this paper we present a lower bound for the fault-diameter of a generalized cycle. Then, we calculate the wide-diameter and the fault-diameter of the families mentioned above by constructing disjoint paths between any pair of vertices. We conclude that the values of these parameters for$BGC(p,d,d^p)$ and $KGC(p,d,d^{p+k}+d^p)$ exceed the lower bound in at most, one unit.
We survey the literature on the eccentricity sequence of a connected graph and make the following contribution. The eccentricity sequence of a graph G is the list of its eccentricities in non-increasing order. Two graphs G1 and G2 are... more
We survey the literature on the eccentricity sequence of a connected graph and make the following contribution. The eccentricity sequence of a graph G is the list of its eccentricities in non-increasing order. Two graphs G1 and G2 are co-eccentric when they have the same eccentricity sequence. Then, we say that G1 and G2 are co-eccentric mates. We characterize the eccentricity sequence of almost all graphs.
Iterated line digraphs have some good properties in relation to the design of interconnection networks. The diameter vulnerability of a digraph is the maximum diameter of the subdigraphs obtained by deleting a ,xed number of vertices or... more
Iterated line digraphs have some good properties in relation to the design of interconnection networks. The diameter vulnerability of a digraph is the maximum diameter of the subdigraphs obtained by deleting a ,xed number of vertices or arcs. This parameter is related to the fault-tolerance of interconnection networks. In this work, we introduce some new parameters in order to ,nd new bounds for the diameter vulnerability of general iterated line digraphs.
Stream ciphering devices seem to be one of the best alternatives in order to provide confidentiality to high-speed transmissions. Several indexes on the security of stream ciphers have been proposed for guaranteeing their strength.... more
Stream ciphering devices seem to be one of the best alternatives in order to provide confidentiality to high-speed transmissions.
Several indexes on the security of stream ciphers have been proposed for guaranteeing their strength. According to the literature, linear complexity of the key stream, randomness and correlation immune attacks are of great importance. But the proposed indexes are not sufficient to guarantee the security of stream ciphers. It is possible that sequences with a high linear complexity have a very bad “linear complexity stability”, i.e., after changing a few bits of the original sequence, its linear complexity decreases or increases fast. In this case, a BAA attack (Best Affine Approximation) could be very successful; the sequences may be very well approximated by another one with very lower linear unpredictability. This problem is especially important when linear feedback shift registers are used. To solve this problem, non-linear next state functions are the possible solution. This paper shows some techniques in order to analyze non-linear functions with maximum period and to set necessary conditions.
Several indexes on the security of stream ciphers have been proposed for guaranteeing their strength. According to the literature, linear complexity of the key stream, randomness and correlation immune attacks are of great importance. But the proposed indexes are not sufficient to guarantee the security of stream ciphers. It is possible that sequences with a high linear complexity have a very bad “linear complexity stability”, i.e., after changing a few bits of the original sequence, its linear complexity decreases or increases fast. In this case, a BAA attack (Best Affine Approximation) could be very successful; the sequences may be very well approximated by another one with very lower linear unpredictability. This problem is especially important when linear feedback shift registers are used. To solve this problem, non-linear next state functions are the possible solution. This paper shows some techniques in order to analyze non-linear functions with maximum period and to set necessary conditions.
Research Interests:
The National Council of Teachers of Mathematics’ [NCTM] Principles and Standards for School Mathematics (2000) states that the Equity Principle means making challenging mathematics accessible to all students. This includes fostering... more
The National Council of Teachers of Mathematics’ [NCTM] Principles and Standards for School Mathematics (2000) states that the Equity Principle means making challenging mathematics accessible to all students. This includes fostering students’ outstanding talent and/or interests in higher mathematics. The purpose of this article is to discuss a method for providing high school students an arena for the study of advanced mathematics through summer mathematics research projects [SMRP].