Narrow Results

Let $H=(V,E)$ be an undirected, planar, non-trivial, connected, and simple graph, where the sets $V$ and $E$ are respectively the vertex and edge sets for $H$. The cardinalities of the sets $E$ and $V$ (i.e., $\left|E\right|$ and $\left|V\right|$) are referred to be the size and order of $H$. A subset $J=\{{\alpha }_1,{\alpha }_2,{\alpha }_3,\dots ,{\alpha }_p\}$ (ordered vertices) of $V$ is termed as a resolving set [edge resolving set (ERS)] for $H$ if for every two distinct vertices $v_i,v_j\in V(H)$ [edges $e_i,e_j\in E(H)$], there is a vertex ${\alpha }_i\in J$ such that $d({\alpha }_i,v_i)\ne d({\alpha }_i,v_j)$ [$d({\alpha }_i,e_i)\ne d({\alpha }_i,e_j)$]. A resolving set (ERS) consisting of the minimum number of vertices is called the metric basis [edge metric basis (EMB)] for $H$ and the cardinality of metric basis (EMB) set is its metric dimension (edge metric dimension), represented by $\mathrm{\text{dim}}(H)$ [$\mathrm{\text{edim}}(H)$]. In this paper, we initiate the study of edge and vertex resolvability parameters for a novel class of planar graphs. Further, we will show that the planar graph possesses an independent minimum vertex and the edge resolving sets having cardinality 3 or 4.

In this paper, we define a Jordan totient graph as an undirected graph with a vertex set $V$ consisting of first $q^2-1$ natural numbers, where $q$ is an odd prime, and an edge set $E$ consisting of a pair of distinct nodes with the property that these two along with n have no common divisor greater than one. Our investigation focuses on three primary objectives. First, we examine the applicability of standard graph-theoretic concepts to Jordan totient graphs, specifically exploring instances where the graph parameters $k$ and $n=q^2$ are derived from the Jordan totient function $J_k(n)$. Second, we develop a combinatorial formula to enumerate the number of triangles within these graphs for given $k$ and $n$. Finally, we propose a model for scalable free social networks using Jordan totient graphs, differentiating between two groups of entities: one with internal isolation but external connections, and the other with unrestricted relationships.

A version of Sharkovskii's cycle coexistence theorem is formulated for a composition of connectivity G_δ-relations with closed values. Thus, a multivalued version in [Andres & Pastor, 2005] holding with at most two exceptions for M-maps, jointly with a single-valued version in [Szuca, 2003], for functions with a connectivity G_δ-graph, are generalized. In particular, our statement is applicable to differential inclusions as well as to some discontinuous functions.

The commuting graph of a group G, denoted by Γ(G), is a simple undirected graph whose vertices are all non-central elements of G and two distinct vertices x, y are adjacent if xy = yx. The commuting graph of a subset of a group is defined similarly. In this paper we investigate the properties of the commuting graph of the symmetric and alternating and subsets of transpositions and involutions in the symmetric groups.

Let R be a commutative ring with nonzero identity and H be a nonempty proper subset of R such that R\H is a saturated multiplicatively closed subset of R. The generalized total graph of R is the (simple) graph GT_H(R) with all elements of R as the vertices, and two distinct vertices x and y are adjacent if and only if x + y ∈ H. In this paper, we investigate the structure of GT_H(R).

In this paper, we define and study the annihilator graph associated to a commutative semigroup with zero. Also, we characterize all annihilator graphs with three vertices.

Let $G$ be a finite non-abelian group and $Z(G)$ be its center. For a fixed nonidentity element $g$ of $G$, the $g$-noncommuting graph of $G$, denoted by ${\mathrm{\Delta }}_G^g$, is a simple undirected graph in which its vertices are $G\setminus Z(G)$ and two distinct vertices $x$ and $y$ are adjacent if $[x,y]\ne g$ and $g^{-1}$. In this paper, we discuss about connectivity of ${\mathrm{\Delta }}_G^g$ and determine all finite non-abelian groups such that their $g$-noncommuting graphs are 1-planar, toroidal or projective.

The power graph of a group $G$ is the simple graph ${𝒫}(G)$, with vertex-set $G$ and vertices $x$ and $y$ are adjacent, if and only if $x\ne y$ and either $y=x^m$ or $x=y^m$ for some positive integer $m$. The proper power graph of $G$, denoted ${{𝒫}}^{\ast }(G)$, is the graph obtained from ${𝒫}(G)$ by deleting the vertex $1$. In [On the connectivity of proper power graphs of finite groups, Comm. Algebra43 (2015) 4305–4319], it is proved that if $n\ge 9$ and neither $n$ nor $n-1$ is a prime, then ${{𝒫}}^{\ast }(S_n)$ is connected and $d({{𝒫}}^{\ast }(S_n))\le 26$. In this paper, we improve the diameter bound of ${{𝒫}}^{\ast }(S_n)$ for which ${{𝒫}}^{\ast }(S_n)$ is connected. We show that $d({{𝒫}}^{\ast }(S_9))=d({{𝒫}}^{\ast }(S_{10}))=14$, $d({{𝒫}}^{\ast }(S_{15}))=12$, and $6\le d({{𝒫}}^{\ast }(S_n))\le 10$ for $n\ge 16$. We also describe a number of short paths in these power graphs.

Let $R$ be a ring with nonzero identity. The coidempotent graph of $R$, denoted it by $G_{I{d}^{-}}(R)$, has its set of vertices equal to the set of all elements of $R$; distinct vertices $x,y$ are adjacent in $G_{I{d}^{-}}(R)$ if and only if $x-y$ or $y-x$ is an idempotent of $R$. In this paper, we study some basic properties of $G_{I{d}^{-}}(R)$ and find some results about the ring-theoretic properties of $R$ and graph-theoretic properties of $G_{I{d}^{-}}(R)$.

Evolutionary studies have been of prime importance to life scientists since ancient times. The advancements in technology has made it possible to make available the massive amounts of genomic data. The abundance of genomic data poses new challenges for biologists, computer scientists and mathematicians to develop approaches for discovery of new relationships in data and evolutionary networks. In this work, nucleotide sequences are converted into binary sequences to explore the network among different species. A new approach based on binary sequences has been proposed to reconstruct the accurate phylogenetic network. The algorithm developed is validated by comparing the results with those obtained by already existing method of network construction. A program is also coded in C language on the Intel Core i3 Dell inspiron machine to obtain the evolutionary network. The new approach developed also provides the fast solutions as there is no need of aligning the sequences.

This paper considers the finite-time consensus problem for a stochastic multi-species system. First, we give out a nonlinear consensus protocol for the multi-species system with Brownian motion, and propose the definition of finite-time consensus in probability. Second, we prove that the multi-species system can achieve finite-time consensus in probability with different proper protocols by use of graph theory, stochastic Lyapunov function method and probability theory. Finally, some simulations are provided to illustrate the effectiveness of the theoretical results.

For a simple and connected graph $G$ with $n$ vertices and the minimum degree two, we show that $H(G)\ge 4+\frac{1}{n-1}-\frac{12}{n+1}$ by a technique based on linear programming, where $H(G)$ is the harmonic index of a graph $G$, defined as the sum of the weights $\frac{2}{d_u+d_v}$ of all edges $uv$ of $G$, $d_u$ denotes the degree of a vertex $u$, and characterize the graph with the minimum value.

Let $A$ be a nonzero commutative ring with identity. Let $n$ be a positive integer. Let $R=A\times \cdots \times A$ ($n$ times). The total dot product graph of $R$ denoted by $\mathrm{\text{TD}}(R)$ is an undirected graph with vertex set which equals $R\setminus \{(0,\dots ,0)\}$ and distinct vertices $x$ and $y$ are adjacent if and only if $x\cdot y=0$, where $x\cdot y$ is the normal dot product of $x$ and $y$. Let $Z(R)$ denote the set of all zero-divisors of $R$ and let us denote $Z(R)\setminus \{(0,\dots ,0)\}$ by $Z{(R)}^{*}$. The zero-divisor dot product graph of $R$, denoted by $\mathrm{\text{ZD}}(R)$ is the subgraph of $\mathrm{\text{TD}}(R)$ induced by $Z{(R)}^{*}$. The graph $\mathrm{\text{TD}}(R)$ (respectively, $\mathrm{\text{ZD}}(R)$) was introduced and investigated by Badawi [Commun. Algebra 43(1) (2015) 43–50]. In this paper, we characterize $R$ such that ${(\mathrm{\text{TD}}(R))}^c$ (respectively, ${(\mathrm{\text{ZD}}(R))}^c$) is connected and determine the diameter and the radius of ${(\mathrm{\text{TD}}(R))}^c$ (respectively, ${(\mathrm{\text{ZD}}(R))}^c$) whenever it is connected. Moreover, if ${(\mathrm{\text{TD}}(R))}^c$ (respectively, ${(\mathrm{\text{ZD}}(R))}^c$) is connected, then we characterize $R$ such that ${(\mathrm{\text{TD}}(R))}^c$ (respectively, ${(\mathrm{\text{ZD}}(R))}^c$) admits a cut vertex.

Some concepts in graph theory are resolving set, dominating set, and dominant metric dimension. A resolving set of a connected graph $G$ is the ordered set $W\subseteq V(G)$ such that every pair of two vertices $u,v\in V(G)$ has the different representation with respect to $W$. A Dominating set of $G$ is the subset $S\subseteq V(G)$ such that for every vertex $x$ in $V(G)\setminus S$ is adjacent to at least one vertex in $S$. A dominant resolving set of $G$ is an ordered set $W\subseteq V(G)$ such that $W$ is a resolving set and a dominating set of $G$. The minimum cardinality of a dominant resolving set is called a dominant metric dimension of $G$, denoted by $D\mathrm{\text{dim}}(G)$. In this paper, we determine the dominant metric dimension of the joint product graphs.