An automorphism of a graph describes its structural symmetry and the concept of fixing number of ... more An automorphism of a graph describes its structural symmetry and the concept of fixing number of a graph is used for breaking its symmetries (except the trivial one). In this paper, we evaluate automorphisms of the co-normal product graph G 1 * G 2 of two simple graphs G 1 and G 2 and give sharp bounds on the order of its automorphism group. We study the fixing number of G 1 * G 2 and prove sharp bounds on it. Moreover, we compute the fixing number of the co-normal product of some families of graphs.
In this paper, we investigate the problem of covering the vertices of a graph associated to a fin... more In this paper, we investigate the problem of covering the vertices of a graph associated to a finite vector space as introduced by Das [16], such that we can uniquely identify any vertex by examining the vertices that cover it. We use locating-dominating sets and identifying codes, which are closely related concepts for this purpose. These sets consist of a dominating set of graph such that every vertex is uniquely identified by its neighborhood within the dominating sets. We find the location-domination number and the identifying number of the graph and study the exchange property for locating-dominating sets and identifying codes.
UDC 512.5 We study the problem of covering of the vertices of a graph associated with a finite ve... more UDC 512.5 We study the problem of covering of the vertices of a graph associated with a finite vector space introduced by Das [Comm. Algebra, 44, 3918-3926 (2016)] such that any vertex can be uniquely identified by examining the vertices from the covering. For this purpose, we use the locating-dominating sets and the identifying codes, i.e., closely related concepts. We find the location-domination number and the identifying number of the graph and study the exchange property of locating-dominating sets and identifying codes.
A non-zero component graph G(V) associated to a finite vector space V is a graph whose vertices a... more A non-zero component graph G(V) associated to a finite vector space V is a graph whose vertices are non-zero vectors of V and two vertices are adjacent, if their corresponding vectors have at least one non-zero component common in their linear combination of basis vectors. In this paper, we extend the study of properties of automorphisms of non-zero component graphs. We prove that every permutation of basis vectors can be extended to an automorphism of G(V). We prove that the symmetric group of basis vectors of V is isomorphic to the automorphism group of G(V). We find the distinguishing number of the graph for both of the cases, when the number of field elements of vector space V are 2 or more than 2.
Soft set theory and rough set theory are mathematical tools to deal with uncertainties. In [3], a... more Soft set theory and rough set theory are mathematical tools to deal with uncertainties. In [3], authors combined these concepts and introduced soft rough sets. In this paper, we introduce the concepts of soft rough graphs, vertex and edge induced soft rough graphs and soft rough trees. We define some products with examples in soft rough graphs.
In this paper, we discuss automorphism related parameters of a graph associated to a finite vecto... more In this paper, we discuss automorphism related parameters of a graph associated to a finite vector space. The fixing neighborhood of a pair (u,v) of vertices of a graph G is the set of all those vertices w of G, such that the orbits of u and v under the action of stabilizer of w are not equal. The fixed number of a graph is the minimum number k such that every subset of vertices of G of cardinality k is a fixing set of G. We study some properties of automorphisms of a graph associated to finite vector space and find the fixing neighborhood of pair of vertices of the graph. We also find the fixed number of the graph. It is shown that, for every positive integer N, there exists a graph G with fxd(G)-fix(G)≥ N, where fxd(G) is the fixed number and fix(G) is the fixing number of G.
A locating-dominating set of a graph G is a dominating set of G such that every vertex of G outsi... more A locating-dominating set of a graph G is a dominating set of G such that every vertex of G outside the dominating set is uniquely identified by its neighborhood within the dominating set. The location-domination number of G is the minimum cardinality of a locating-dominating set in G. Let G_1 and G_2 be the disjoint copies of a graph G and f:V(G_1)→ V(G_2) be a function. A functigraph F^f_G consists of the vertex set V(G_1)∪ V(G_2) and the edge set E(G_1)∪ E(G_2)∪{uv:v=f(u)}. In this paper, we study the variation of the location-domination number in passing from G to F^f_G and find its sharp lower and upper bounds. We also study the location-domination number of functigraphs of the complete graphs for all possible definitions of the function f. We also obtain the location-domination number of functigraphs of a family of spanning subgraph of the complete graphs.
A hypergraph is a generalization of a graph where edges can connect any number of vertices. In th... more A hypergraph is a generalization of a graph where edges can connect any number of vertices. In this paper, we extend the study of locating-dominating sets to hypergraphs. Along with some basic results, sharp bounds for the location-domination number of hypergraphs in general and exact values with specified conditions are investigated. Moreover, locating-dominating sets in some specific hypergraphs are found.
Let G_1 and G_2 be disjoint copies of a graph G, and let g:V(G_1)→ V(G_2) be a function. A functi... more Let G_1 and G_2 be disjoint copies of a graph G, and let g:V(G_1)→ V(G_2) be a function. A functigraph F_G consists of the vertex set V(G_1)∪ V(G_2) and the edge set E(G_1)∪ E(G_2)∪{uv:g(u)=v}. In this paper, we extend the study of the distinguishing number of a graph to its functigraph. We discuss the behavior of the distinguishing number in passing from G to F_G and find its sharp lower and upper bounds. We also discuss the distinguishing number of functigraphs of complete graphs and join graphs.
In this paper, we investigate the theory of rough set to study graphs using the concept of orbits... more In this paper, we investigate the theory of rough set to study graphs using the concept of orbits. Rough sets are based on a clustering criterion and we use the idea of similarity of vertices under automorphism as a criterion. We introduce indiscernibility relation in terms of orbits and prove necessary and sufficient conditions under which the indiscernibility partitions remain the same when associated with different attribute sets. We show that automorphisms of the graph $ \mathcal{G} $ preserve the indiscernibility partitions. Further, we prove that for any graph $ \mathcal{G} $ with $ k $ orbits, any reduct $ \mathcal{R} $ consists of one element from $ k-1 $ orbits of the graph. We also study the rough membership functions for paths, cycles, complete and complete bipartite graphs. Moreover, we introduce essential sets and discernibility matrices induced by orbits of graphs and study their relationship. We also prove that every essential set consists of union of any two orbits o...
Abstract: A vertex u in a graph G resolves a pair of distinct vertices x, y of G if the distance ... more Abstract: A vertex u in a graph G resolves a pair of distinct vertices x, y of G if the distance between u and x is different from the distance between u and y. A set W of vertices in G resolves the graph G if every pair of distinct vertices of G is resolved by some vertices in W . The metric dimension of a graph, denoted by dim (G), is the smallest cardinality of a resolving set. A resolving set W for a connected graph G of order n ≥ 3 is called 2-size resolving set if the size of the subgraph < W > induced by W is two. The minimum cardinality of a 2-size resolving set is called the 2-size metric dimension of G, denoted by tr(G). A 2-size resolving set of cardinality tr(G) is called a tr-set. In this paper, we study 2-size resolving sets in some well-known classes of graphs and give some realizable results.
We prove Posner’s first theorem as well as second theorem in prime Γ-rings. We also show that if ... more We prove Posner’s first theorem as well as second theorem in prime Γ-rings. We also show that if M is a 2-torsion free semiprime Γ-ring and f : M →M an additive mapping such that 2f(xβx) = f(x)βθ(x) + θ(x)βf(x) for all x ∈M , β ∈ Γ, where θ is a surjective endomorphism of M with θ(Z) = Z, then f is a θ-centralizer.
The fixing number of a graph $G$ is the smallest cardinality of a set of vertices $F\subseteq V(G... more The fixing number of a graph $G$ is the smallest cardinality of a set of vertices $F\subseteq V(G)$ such that only the trivial automorphism of $G$ fixes every vertex in $F$. Let $\Pi$ $=$ $\{F_1,F_2,\ldots,F_k\}$ be an ordered $k$-partition of $V(G)$. Then $\Pi$ is called a {\it fixatic partition} if for all $i$; $1\leq i\leq k$, $F_i$ is a fixing set for $G$. The cardinality of a largest fixatic partition is called the {\it fixatic number} of $G$. In this paper, we study the fixatic numbers of graphs. Sharp bounds for the fixatic number of graphs in general and exact values with specified conditions are given. Some realizable results are also given in this paper.
Let G be a connected graph. For a vertex v ∈ V (G) and an ordered k-partition Π = {S1, S2, ..., S... more Let G be a connected graph. For a vertex v ∈ V (G) and an ordered k-partition Π = {S1, S2, ..., Sk} of V (G), the representation of v with respect to Π is the k-vector r(v|Π) = (d(v, S1), d(v, S2), ..., d(v, Sk)). The k-partition Π is said to be resolving if the k-vectors r(v|Π), v ∈ V (G), are distinct. The minimum k for which there is a resolving k-partition of V (G) is called the partition dimension of G, denoted by pd(G). A resolving k-partition Π = {S1, S2, ..., Sk} of V (G) is said to be connected if each subgraph < Si > induced by Si (1 ≤ i ≤ k) is connected in G. The minimum k for which there is a connected resolving k-partition of V (G) is called the connected partition dimension of G, denoted by cpd(G). In this paper, the partition dimension as well as the connected partition dimension of the wheel Wn with n spokes are considered, by showing that d(2n)e ≤ pd(Wn) ≤ 2dne + 1 and cpd(Wn) = d(n + 2)/3e for ∗The research was done while the third author was visiting School...
The zero forcing number of a graph $G$, denoted by $Z(G)$, is the minimum cardinality of a set $S... more The zero forcing number of a graph $G$, denoted by $Z(G)$, is the minimum cardinality of a set $S$ of black vertices (where vertices in $V(G)\setminus S$ are colored white) such that $V(G)$ is turned black after finitely many applications of $"$the color change rule$"$: a white vertex is turned black if it is the only white neighbor of a black vertex. In this paper, we study the zero forcing number of corona product, $G\odot H$ and lexicographic product, $G\circ H$ of two graphs $G$ and $H$. It is shown that if $G$ and $H$ are connected graphs of order $n_{1}\geq2$ and $n_{2}\geq2$ respectively, then $Z(G\odot ^{k}H)=Z(G\odot ^{k-1}H)+n_{1}(n_{2}+1)^{k-1}Z(H)$, where $G\odot^{k}H=(G\odot^{k-1}H)\odot H$. Also, it is shown that for a connected graph $G$ of order $n\geq 2$ and an arbitrary graph $H$ containing $l\geq 1$ components $H_{1},H_{2}, \cdots,H_{l}$ with $|V(H_{i})|=m_{i}\geq 2$, $1\leq i\leq l$, $(n-1)l+\sum\limits_{i=1}^l m_{i}\leq Z(G\circ H)\leq n(\sum\limits_{i...
In this paper, we investigate the problem of covering the vertices of a graph associated to a fin... more In this paper, we investigate the problem of covering the vertices of a graph associated to a finite vector space as introduced by Das \cite{Das}, such that we can uniquely identify any vertex by examining the vertices that cover it. We use locating-dominating sets and identifying codes, which are closely related concepts for this purpose. These sets consist of a dominating set of graph such that every vertex is uniquely identified by its neighborhood within the dominating sets. We find the location-domination number and the identifying number of the graph and study the exchange property for locating-dominating sets and identifying codes.
For given graphs G and H, the Ramsey number R(G,H) is the least natural number n such that for ev... more For given graphs G and H, the Ramsey number R(G,H) is the least natural number n such that for every graph F of order n the following condition holds: either F contains G or the complement of F contains H. In this paper, we determine the Ramsey number of path Pn versus sunflower graph SFm when n grows at least as a quadratic function of m. In this case R(Pn, SFm) = 3n − 2 if m is odd and 2n + m 2 − 2 otherwise.
An automorphism group of a graph $G$ is the set of all permutations of the vertex set of $G$ that... more An automorphism group of a graph $G$ is the set of all permutations of the vertex set of $G$ that preserve adjacency and non adjacency of vertices in a graph. A fixing set of a graph $G$ is a subset of vertices of $G$ such that only the trivial automorphism fixes every vertex in $S$. Minimum cardinality of a fixing set of $G$ is called the fixing number of $G$. In this article, we define a fractional version of the fixing number of a graph. We formulate the problem of finding the fixing number of a graph as an integer programming problem. It is shown that a relaxation of this problem leads to a linear programming problem and hence to a fractional version of the fixing number of a graph. We also characterize the graphs $G$ with the fractional fixing number $\frac{|V(G)|}{2}$ and the fractional fixing number of some families of graphs is also obtained.
An automorphism of a graph describes its structural symmetry and the concept of fixing number of ... more An automorphism of a graph describes its structural symmetry and the concept of fixing number of a graph is used for breaking its symmetries (except the trivial one). In this paper, we evaluate automorphisms of the co-normal product graph G 1 * G 2 of two simple graphs G 1 and G 2 and give sharp bounds on the order of its automorphism group. We study the fixing number of G 1 * G 2 and prove sharp bounds on it. Moreover, we compute the fixing number of the co-normal product of some families of graphs.
In this paper, we investigate the problem of covering the vertices of a graph associated to a fin... more In this paper, we investigate the problem of covering the vertices of a graph associated to a finite vector space as introduced by Das [16], such that we can uniquely identify any vertex by examining the vertices that cover it. We use locating-dominating sets and identifying codes, which are closely related concepts for this purpose. These sets consist of a dominating set of graph such that every vertex is uniquely identified by its neighborhood within the dominating sets. We find the location-domination number and the identifying number of the graph and study the exchange property for locating-dominating sets and identifying codes.
UDC 512.5 We study the problem of covering of the vertices of a graph associated with a finite ve... more UDC 512.5 We study the problem of covering of the vertices of a graph associated with a finite vector space introduced by Das [Comm. Algebra, 44, 3918-3926 (2016)] such that any vertex can be uniquely identified by examining the vertices from the covering. For this purpose, we use the locating-dominating sets and the identifying codes, i.e., closely related concepts. We find the location-domination number and the identifying number of the graph and study the exchange property of locating-dominating sets and identifying codes.
A non-zero component graph G(V) associated to a finite vector space V is a graph whose vertices a... more A non-zero component graph G(V) associated to a finite vector space V is a graph whose vertices are non-zero vectors of V and two vertices are adjacent, if their corresponding vectors have at least one non-zero component common in their linear combination of basis vectors. In this paper, we extend the study of properties of automorphisms of non-zero component graphs. We prove that every permutation of basis vectors can be extended to an automorphism of G(V). We prove that the symmetric group of basis vectors of V is isomorphic to the automorphism group of G(V). We find the distinguishing number of the graph for both of the cases, when the number of field elements of vector space V are 2 or more than 2.
Soft set theory and rough set theory are mathematical tools to deal with uncertainties. In [3], a... more Soft set theory and rough set theory are mathematical tools to deal with uncertainties. In [3], authors combined these concepts and introduced soft rough sets. In this paper, we introduce the concepts of soft rough graphs, vertex and edge induced soft rough graphs and soft rough trees. We define some products with examples in soft rough graphs.
In this paper, we discuss automorphism related parameters of a graph associated to a finite vecto... more In this paper, we discuss automorphism related parameters of a graph associated to a finite vector space. The fixing neighborhood of a pair (u,v) of vertices of a graph G is the set of all those vertices w of G, such that the orbits of u and v under the action of stabilizer of w are not equal. The fixed number of a graph is the minimum number k such that every subset of vertices of G of cardinality k is a fixing set of G. We study some properties of automorphisms of a graph associated to finite vector space and find the fixing neighborhood of pair of vertices of the graph. We also find the fixed number of the graph. It is shown that, for every positive integer N, there exists a graph G with fxd(G)-fix(G)≥ N, where fxd(G) is the fixed number and fix(G) is the fixing number of G.
A locating-dominating set of a graph G is a dominating set of G such that every vertex of G outsi... more A locating-dominating set of a graph G is a dominating set of G such that every vertex of G outside the dominating set is uniquely identified by its neighborhood within the dominating set. The location-domination number of G is the minimum cardinality of a locating-dominating set in G. Let G_1 and G_2 be the disjoint copies of a graph G and f:V(G_1)→ V(G_2) be a function. A functigraph F^f_G consists of the vertex set V(G_1)∪ V(G_2) and the edge set E(G_1)∪ E(G_2)∪{uv:v=f(u)}. In this paper, we study the variation of the location-domination number in passing from G to F^f_G and find its sharp lower and upper bounds. We also study the location-domination number of functigraphs of the complete graphs for all possible definitions of the function f. We also obtain the location-domination number of functigraphs of a family of spanning subgraph of the complete graphs.
A hypergraph is a generalization of a graph where edges can connect any number of vertices. In th... more A hypergraph is a generalization of a graph where edges can connect any number of vertices. In this paper, we extend the study of locating-dominating sets to hypergraphs. Along with some basic results, sharp bounds for the location-domination number of hypergraphs in general and exact values with specified conditions are investigated. Moreover, locating-dominating sets in some specific hypergraphs are found.
Let G_1 and G_2 be disjoint copies of a graph G, and let g:V(G_1)→ V(G_2) be a function. A functi... more Let G_1 and G_2 be disjoint copies of a graph G, and let g:V(G_1)→ V(G_2) be a function. A functigraph F_G consists of the vertex set V(G_1)∪ V(G_2) and the edge set E(G_1)∪ E(G_2)∪{uv:g(u)=v}. In this paper, we extend the study of the distinguishing number of a graph to its functigraph. We discuss the behavior of the distinguishing number in passing from G to F_G and find its sharp lower and upper bounds. We also discuss the distinguishing number of functigraphs of complete graphs and join graphs.
In this paper, we investigate the theory of rough set to study graphs using the concept of orbits... more In this paper, we investigate the theory of rough set to study graphs using the concept of orbits. Rough sets are based on a clustering criterion and we use the idea of similarity of vertices under automorphism as a criterion. We introduce indiscernibility relation in terms of orbits and prove necessary and sufficient conditions under which the indiscernibility partitions remain the same when associated with different attribute sets. We show that automorphisms of the graph $ \mathcal{G} $ preserve the indiscernibility partitions. Further, we prove that for any graph $ \mathcal{G} $ with $ k $ orbits, any reduct $ \mathcal{R} $ consists of one element from $ k-1 $ orbits of the graph. We also study the rough membership functions for paths, cycles, complete and complete bipartite graphs. Moreover, we introduce essential sets and discernibility matrices induced by orbits of graphs and study their relationship. We also prove that every essential set consists of union of any two orbits o...
Abstract: A vertex u in a graph G resolves a pair of distinct vertices x, y of G if the distance ... more Abstract: A vertex u in a graph G resolves a pair of distinct vertices x, y of G if the distance between u and x is different from the distance between u and y. A set W of vertices in G resolves the graph G if every pair of distinct vertices of G is resolved by some vertices in W . The metric dimension of a graph, denoted by dim (G), is the smallest cardinality of a resolving set. A resolving set W for a connected graph G of order n ≥ 3 is called 2-size resolving set if the size of the subgraph < W > induced by W is two. The minimum cardinality of a 2-size resolving set is called the 2-size metric dimension of G, denoted by tr(G). A 2-size resolving set of cardinality tr(G) is called a tr-set. In this paper, we study 2-size resolving sets in some well-known classes of graphs and give some realizable results.
We prove Posner’s first theorem as well as second theorem in prime Γ-rings. We also show that if ... more We prove Posner’s first theorem as well as second theorem in prime Γ-rings. We also show that if M is a 2-torsion free semiprime Γ-ring and f : M →M an additive mapping such that 2f(xβx) = f(x)βθ(x) + θ(x)βf(x) for all x ∈M , β ∈ Γ, where θ is a surjective endomorphism of M with θ(Z) = Z, then f is a θ-centralizer.
The fixing number of a graph $G$ is the smallest cardinality of a set of vertices $F\subseteq V(G... more The fixing number of a graph $G$ is the smallest cardinality of a set of vertices $F\subseteq V(G)$ such that only the trivial automorphism of $G$ fixes every vertex in $F$. Let $\Pi$ $=$ $\{F_1,F_2,\ldots,F_k\}$ be an ordered $k$-partition of $V(G)$. Then $\Pi$ is called a {\it fixatic partition} if for all $i$; $1\leq i\leq k$, $F_i$ is a fixing set for $G$. The cardinality of a largest fixatic partition is called the {\it fixatic number} of $G$. In this paper, we study the fixatic numbers of graphs. Sharp bounds for the fixatic number of graphs in general and exact values with specified conditions are given. Some realizable results are also given in this paper.
Let G be a connected graph. For a vertex v ∈ V (G) and an ordered k-partition Π = {S1, S2, ..., S... more Let G be a connected graph. For a vertex v ∈ V (G) and an ordered k-partition Π = {S1, S2, ..., Sk} of V (G), the representation of v with respect to Π is the k-vector r(v|Π) = (d(v, S1), d(v, S2), ..., d(v, Sk)). The k-partition Π is said to be resolving if the k-vectors r(v|Π), v ∈ V (G), are distinct. The minimum k for which there is a resolving k-partition of V (G) is called the partition dimension of G, denoted by pd(G). A resolving k-partition Π = {S1, S2, ..., Sk} of V (G) is said to be connected if each subgraph < Si > induced by Si (1 ≤ i ≤ k) is connected in G. The minimum k for which there is a connected resolving k-partition of V (G) is called the connected partition dimension of G, denoted by cpd(G). In this paper, the partition dimension as well as the connected partition dimension of the wheel Wn with n spokes are considered, by showing that d(2n)e ≤ pd(Wn) ≤ 2dne + 1 and cpd(Wn) = d(n + 2)/3e for ∗The research was done while the third author was visiting School...
The zero forcing number of a graph $G$, denoted by $Z(G)$, is the minimum cardinality of a set $S... more The zero forcing number of a graph $G$, denoted by $Z(G)$, is the minimum cardinality of a set $S$ of black vertices (where vertices in $V(G)\setminus S$ are colored white) such that $V(G)$ is turned black after finitely many applications of $"$the color change rule$"$: a white vertex is turned black if it is the only white neighbor of a black vertex. In this paper, we study the zero forcing number of corona product, $G\odot H$ and lexicographic product, $G\circ H$ of two graphs $G$ and $H$. It is shown that if $G$ and $H$ are connected graphs of order $n_{1}\geq2$ and $n_{2}\geq2$ respectively, then $Z(G\odot ^{k}H)=Z(G\odot ^{k-1}H)+n_{1}(n_{2}+1)^{k-1}Z(H)$, where $G\odot^{k}H=(G\odot^{k-1}H)\odot H$. Also, it is shown that for a connected graph $G$ of order $n\geq 2$ and an arbitrary graph $H$ containing $l\geq 1$ components $H_{1},H_{2}, \cdots,H_{l}$ with $|V(H_{i})|=m_{i}\geq 2$, $1\leq i\leq l$, $(n-1)l+\sum\limits_{i=1}^l m_{i}\leq Z(G\circ H)\leq n(\sum\limits_{i...
In this paper, we investigate the problem of covering the vertices of a graph associated to a fin... more In this paper, we investigate the problem of covering the vertices of a graph associated to a finite vector space as introduced by Das \cite{Das}, such that we can uniquely identify any vertex by examining the vertices that cover it. We use locating-dominating sets and identifying codes, which are closely related concepts for this purpose. These sets consist of a dominating set of graph such that every vertex is uniquely identified by its neighborhood within the dominating sets. We find the location-domination number and the identifying number of the graph and study the exchange property for locating-dominating sets and identifying codes.
For given graphs G and H, the Ramsey number R(G,H) is the least natural number n such that for ev... more For given graphs G and H, the Ramsey number R(G,H) is the least natural number n such that for every graph F of order n the following condition holds: either F contains G or the complement of F contains H. In this paper, we determine the Ramsey number of path Pn versus sunflower graph SFm when n grows at least as a quadratic function of m. In this case R(Pn, SFm) = 3n − 2 if m is odd and 2n + m 2 − 2 otherwise.
An automorphism group of a graph $G$ is the set of all permutations of the vertex set of $G$ that... more An automorphism group of a graph $G$ is the set of all permutations of the vertex set of $G$ that preserve adjacency and non adjacency of vertices in a graph. A fixing set of a graph $G$ is a subset of vertices of $G$ such that only the trivial automorphism fixes every vertex in $S$. Minimum cardinality of a fixing set of $G$ is called the fixing number of $G$. In this article, we define a fractional version of the fixing number of a graph. We formulate the problem of finding the fixing number of a graph as an integer programming problem. It is shown that a relaxation of this problem leads to a linear programming problem and hence to a fractional version of the fixing number of a graph. We also characterize the graphs $G$ with the fractional fixing number $\frac{|V(G)|}{2}$ and the fractional fixing number of some families of graphs is also obtained.
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