The pairwise balanced designs (PB_D's) is a pair (P, B), where P is a finite set of v points and В is a family of subsets of P, called blocks such that every two distinct points in P appear in ...exactly one block. Let a(G) = a and /3(G) = ß be the vertex covering and independence number of a graph G = (V, B) with the minimum and maximum cardinality of such sets are denoted by а-sets and /3-sets of G, respectively (or, simply (a,/3)-sets). In this paper, we obtain the total number of (a, /3)-sets in different jump sizes of some circulant graphs apart from strongly regular graphs which are the blocks of PBD.
Recent papers have formulated the problem of learning graphs from data as an inverse covariance estimation problem with graph Laplacian constraints. While such problems are convex, existing methods ...cannot guarantee that solutions will have specific graph topology properties (e.g., being a tree), which are desirable for some applications. The problem of learning a graph with topology properties is in general non-convex. In this paper, we propose an approach to solve these problems by decomposing them into two sub-problems for which efficient solutions are known. Specifically, a graph topology inference (GTI) step is employed to select a feasible graph topology. Then, a graph weight estimation (GWE) step is performed by solving a generalized graph Laplacian estimation problem, where edges are constrained by the topology found in the GTI step. Our main result is a bound on the error of the GWE step as a function of the error in the GTI step. This error bound indicates that the GTI step should be solved using an algorithm that approximates the data similarity matrix by another matrix whose entries have been thresholded to zero to have the desired type of graph topology. The GTI stage can leverage existing methods, which are typically based on minimizing the total weight of removed edges. Since the GWE stage is an inverse covariance estimation problem with linear constraints, it can be solved using existing convex optimization methods. We demonstrate that our approach can achieve good results for both synthetic and texture image data.
Strong product of two S + -valued graphs Sundar, M.; Chandramouleeswaran, M.
IOP conference series. Materials Science and Engineering,
03/2021, Volume:
1084, Issue:
1
Journal Article
Peer reviewed
Open access
Abstract
Begun in 2015, followed by S-valued graphs, we study the strong product of two S
+
-valued graphs
G
S
1
and
G
s
2
and the generalized strong product of two S
+
valued graphs
G
S
1
and
G
s
2
.... Furthermore, we contend the regularity conditions on generalized strong product of S
+
-valued graphs.
A COMPREHENSIVE APPROACH TO GEOMETRIC SIMPLEXES Mohammed, Mari; Obeng-Denteh, William; Asante-Mensa, Fred
Global journal of pure and applied sciences,
01/2022, Volume:
28, Issue:
1
Journal Article
This paper is an expository research to Simplexes. The work focuses on how simplexes are created. It also looked at how complete graphs are treated as simplexes. We further present an important ...theorem and its proof.
Graphs Defined on Rings: A Review Madhumitha, S.; Naduvath, Sudev
Mathematics (Basel),
09/2023, Volume:
11, Issue:
17
Journal Article
Peer reviewed
Open access
The study on graphs emerging from different algebraic structures such as groups, rings, fields, vector spaces, etc. is a prominent area of research in mathematics, as algebra and graph theory are two ...mathematical fields that focus on creating and analysing structures. There are numerous studies linking algebraic structures and graphs, which began with the introduction of Cayley graphs of groups. Several algebraic graphs have been defined on rings, a fast-growing area in the literature. In this article, we systematically review the literature on some variants of Cayley graphs that are defined on rings and highlight the properties and characteristics of such graphs, to showcase the research in this area.
Let G be a graph with p vertices and q edges and an injective function where each is a odd Fibonacci number and the induced edge labeling are defined by and all these edge labeling are distinct is ...called Odd Fibonacci Stolarsky-3 Mean Labeling. A graph which admits a Odd Fibonacci Stolarsky-3 Mean Labeling is called a Odd Fibonacci Stolarsky-3 mean graph.
Counting short cycles in bipartite graphs is a fundamental problem of interest in the analysis and design of low-density parity-check codes. The vast majority of research in this area is focused on ...algorithmic techniques. Most recently, Blake and Lin proposed a computational technique to count the number of cycles of length <inline-formula> <tex-math notation="LaTeX">\boldsymbol {g} </tex-math></inline-formula> in a bi-regular bipartite graph, where <inline-formula> <tex-math notation="LaTeX">\boldsymbol {g} </tex-math></inline-formula> is the girth of the graph. The information required for the computation is the node degree and the multiplicity of the nodes on both sides of the partition, as well as the eigenvalues of the adjacency matrix of the graph (graph spectrum). In this paper, the result of Blake and Lin is extended to compute the number of cycles of length <inline-formula> <tex-math notation="LaTeX">\boldsymbol {g} + \textbf {2}, \ldots, \textbf {2}\boldsymbol {g}-\textbf {2} </tex-math></inline-formula>, for bi-regular bipartite graphs, as well as the number of 4-cycles and 6-cycles in irregular and half-regular bipartite graphs, with <inline-formula> <tex-math notation="LaTeX">\boldsymbol {g} \geq \textbf {4} </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">\boldsymbol {g} \geq \textbf {6} </tex-math></inline-formula>, respectively.
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