Let $R$ be a commutative ring with identity and $I$ a proper ideal of $R$. In this paper we intro... more Let $R$ be a commutative ring with identity and $I$ a proper ideal of $R$. In this paper we introduce the ideal-based quasi zero divisor graph $Q\Gamma_{I}(R)$ of $R$ with respect to $I$ which is an undirected graph with vertex set $V=\{a\in R\backslash\sqrt{I}:$ $ab\in I$ for some $b\in R\backslash\sqrt{I}\}$ and two distinct vertices $a$ and $b$ are adjacent if and only if $ab\in I$. We study the basic properties of this graph such as diameter, girth, dominaton number, etc. We also investigate the interplay between the ring theoretic properties of a Noetherian multiplication ring $R$ and the graph-theoretic properties of $Q\Gamma_{I}(R)$.
Discussiones Mathematicae - General Algebra and Applications
In this paper, we introduce a new family of circulants GA(t, k), called Generalized Andrásfai gra... more In this paper, we introduce a new family of circulants GA(t, k), called Generalized Andrásfai graphs, where t, k ≥ 2 are integers. We study various parameters like diameter, girth, domination number etc. of GA(t, k). Moreover, we find the full automorphism group of GA(t, k) and compute its determining number.
The difference graph D(G) of a finite group G is the difference of enhanced power graph of G and ... more The difference graph D(G) of a finite group G is the difference of enhanced power graph of G and power graph of G, with all isolated vertices are removed. In this paper we study the connectedness and perfectness of D(G) with respect to various properties of the underlying group G. We also find several connection between the difference graph of G and the Gruenberg-Kegel graph of G.
Let [Formula: see text] be a commutative ring with identity. The prime ideal sum graph of [Formul... more Let [Formula: see text] be a commutative ring with identity. The prime ideal sum graph of [Formula: see text], denoted by [Formula: see text], is a graph whose vertices are nonzero proper ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is a prime ideal of [Formula: see text]. In this paper, we study some connections between the graph-theoretic properties of this graph and some algebraic properties of rings. The clique number, the chromatic number and the domination number of the prime ideal sum graph for some classes of rings are studied. It is observed that under which condition [Formula: see text] is complete. Moreover, the diameter and the girth of [Formula: see text] are studied.
From its inception, public-key cryptosystems have been an area of active research. Various aspect... more From its inception, public-key cryptosystems have been an area of active research. Various aspects of public-key encryption like constructions, security notions, adversarial models, hardness assumptions, proof-methodology, efficiency, compatibility etc. have been analysed and re-analysed in the last three and half decades by numerous cryptographers. Some of them are good enough to survive while some of them, though broken, provides meaningful insights towards the subject. In this article, our aim is to provide an expository as well as technical (as far as possible, keeping in mind its brevity) overview of the subject as it has progressed over the years, along with some open problems and suitable references.
Electronic Journal of Graph Theory and Applications
In this paper, we introduce a new family of cubic graphs Γ(m), called Generalized Pappus graphs, ... more In this paper, we introduce a new family of cubic graphs Γ(m), called Generalized Pappus graphs, where m ≥ 3. We compute the automorphism group of Γ(m) and characterize when it is a Cayley graph.
Discrete Mathematics & Theoretical Computer Science, Mar 30, 2022
The determining number of a graph G = (V, E) is the minimum cardinality of a set S ⊆ V such that ... more The determining number of a graph G = (V, E) is the minimum cardinality of a set S ⊆ V such that pointwise stabilizer of S under the action of Aut(G) is trivial. In this paper, we provide some improved upper and lower bounds on the determining number of Kneser graphs. Moreover, we provide the exact value of the determining number for some subfamilies of Kneser graphs.
In this paper, we introduce a new graph, called subgroup product graph SP (G) of a group G. The s... more In this paper, we introduce a new graph, called subgroup product graph SP (G) of a group G. The set of vertices of SP (G) is the set of all non-trivial proper subgroups of G and two vertices H and K are adjacent if and only if HK = G. We characterize when SP (G) is connected, a complete graph, star graph, has an universal vertex etc. We also find various graph parameters like diameter, girth, bipartiteness etc. in terms of properties of G.
Rose window graphs are a family of tetravalent graphs, introduced by Steve Wilson. Following it, ... more Rose window graphs are a family of tetravalent graphs, introduced by Steve Wilson. Following it, Kovacs, Kutnar and Marusic classified the edge-transitive rose window graphs and Dobson, Kovacs and Miklavic characterized the vertex transitive rose window graphs. In this paper, we classify the Cayley rose window graphs.
The determining number of a graph $G = (V,E)$ is the minimum cardinality of a set $S\subseteq V$ ... more The determining number of a graph $G = (V,E)$ is the minimum cardinality of a set $S\subseteq V$ such that pointwise stabilizer of $S$ under the action of $Aut(G)$ is trivial. In this paper, we prove some improved upper and lower bounds on the determining number of Kneser graphs. Moreover, we compute the exact value of the determining number for some subfamilies of Kneser graphs. Finally, we show that the number of Kneser graphs with a given determining number $r$ is an increasing function of $r$.
Discrete Mathematics, Algorithms and Applications, 2021
The neighborhood graph [Formula: see text] of a graph [Formula: see text] has the same vertex set... more The neighborhood graph [Formula: see text] of a graph [Formula: see text] has the same vertex set as [Formula: see text] and two vertices are adjacent in [Formula: see text] if and only if they have a common neighbor in [Formula: see text]. We study the domination number, independence number and chromatic number of [Formula: see text] in terms of those of [Formula: see text]. Moreover we investigate various structural similarities between [Formula: see text] and [Formula: see text].
In this paper, we initiate the study of Paley-type graphs \(\Gamma_N\) modulo \(N=pq\), where \(p... more In this paper, we initiate the study of Paley-type graphs \(\Gamma_N\) modulo \(N=pq\), where \(p,q\) are distinct primes of the form \(4k+1\). It is shown that \(\Gamma_N\) is an edge-regular, symmetric, Eulerian and Hamiltonian graph. Also, the vertex connectivity, edge connectivity, diameter and girth of \(\Gamma_N\) are studied and their relationship with the forms of \(p\) and \(q\) are discussed. Moreover, we specify the forms of primes for which \(\Gamma_N\) is triangulated or triangle-free and provide some bounds (exact values in some particular cases) for the order of the automorphism group \(\operatorname{Aut}(\Gamma_N)\) of the graph \(\Gamma_N\), the chromatic number, the independence number, and the domination number of \(\Gamma_N\).
In this paper, we introduce the notion of connected fair domination in graphs. A connected fair d... more In this paper, we introduce the notion of connected fair domination in graphs. A connected fair dominating set in a graph G (or \(\mathsf {CFD}\)-set) is a dominating set S such that \(\langle S \rangle \) is connected in G and all vertices not in S are dominated by the same number of vertices from S, i.e., every two vertices not in S has the same number of neighbours in S. The connected fair domination number of G (\(\mathsf {cfd}(G)\)) is the minimum cardinality of a \(\mathsf {CFD}\)-set in G. Apart from finding \(\mathsf {cfd}(G)\) for some standard graphs G, we proved various bounds on \(\mathsf {cfd}(G)\) in terms of order and some other graph parameters of G.
The annihilating-ideal graph of a commutative ring R with unity is defined as the graph AG(R) wit... more The annihilating-ideal graph of a commutative ring R with unity is defined as the graph AG(R) with the vertex set is the set of all non-zero ideals with non-zero annihilators and two distinct vertices I and J are adjacent if and only if IJ = 0. Nikandish et.al. proved that AG(Zn) is weakly perfect. In this short paper, we characterize n for which AG(Zn) is perfect.
In this paper, we introduce a graph structure, called zero-set intersection graph Γ(C(X)), on the... more In this paper, we introduce a graph structure, called zero-set intersection graph Γ(C(X)), on the ring of real valued continuous functions, C(X), on a Tychonoff space X. We show that the graph is connected and triangulated. We also study the inter-relationship of cliques of Γ(C(X)) and ideals in C(X) which helps to characterize the structure of maximal cliques of Γ(C(X)) by different kind of maximal ideals of C(X). We show that there are at least 2c many different maximal cliques which are never graph isomorphic to each other. Furthermore, we study the neighbourhood properties of a vertex and show its connection with the topology of X and algebraic properties of C(X). Finally, it is shown that two graphs are isomorphic if and only if the corresponding rings are isomorphic if and only if the corresponding topologies are homeomorphic either for first countable topological spaces or for realcompact topological spaces.
The co-maximal subgroup graph Γ(G) of a group G is a graph whose vertices are nontrivial proper s... more The co-maximal subgroup graph Γ(G) of a group G is a graph whose vertices are nontrivial proper subgroups of G and two vertices H and K are adjacent if HK = G. In this paper, we continue the study of Γ(G), especially when Γ(G) has isolated vertices. We define a new graph Γ∗(G), which is obtained by removing isolated vertices from Γ(G). We characterize when Γ∗(G) is connected, a complete graph, star graph, has an universal vertex etc. We also find various graph parameters like diameter, girth, bipartiteness etc. in terms of properties of G.
In this paper, we introduce a new family of graphs, $\Gamma(n,a)$. We show that it is an infinite... more In this paper, we introduce a new family of graphs, $\Gamma(n,a)$. We show that it is an infinite family of tetravalent half-transitive Cayley graphs. Apart from that, we determine some structural properties of $\Gamma(n,a)$.
In this short paper, we characterize the positive integers n for which intersection graph of idea... more In this short paper, we characterize the positive integers n for which intersection graph of ideals of ℤn is perfect.
Let $R$ be a commutative ring with identity and $I$ a proper ideal of $R$. In this paper we intro... more Let $R$ be a commutative ring with identity and $I$ a proper ideal of $R$. In this paper we introduce the ideal-based quasi zero divisor graph $Q\Gamma_{I}(R)$ of $R$ with respect to $I$ which is an undirected graph with vertex set $V=\{a\in R\backslash\sqrt{I}:$ $ab\in I$ for some $b\in R\backslash\sqrt{I}\}$ and two distinct vertices $a$ and $b$ are adjacent if and only if $ab\in I$. We study the basic properties of this graph such as diameter, girth, dominaton number, etc. We also investigate the interplay between the ring theoretic properties of a Noetherian multiplication ring $R$ and the graph-theoretic properties of $Q\Gamma_{I}(R)$.
Discussiones Mathematicae - General Algebra and Applications
In this paper, we introduce a new family of circulants GA(t, k), called Generalized Andrásfai gra... more In this paper, we introduce a new family of circulants GA(t, k), called Generalized Andrásfai graphs, where t, k ≥ 2 are integers. We study various parameters like diameter, girth, domination number etc. of GA(t, k). Moreover, we find the full automorphism group of GA(t, k) and compute its determining number.
The difference graph D(G) of a finite group G is the difference of enhanced power graph of G and ... more The difference graph D(G) of a finite group G is the difference of enhanced power graph of G and power graph of G, with all isolated vertices are removed. In this paper we study the connectedness and perfectness of D(G) with respect to various properties of the underlying group G. We also find several connection between the difference graph of G and the Gruenberg-Kegel graph of G.
Let [Formula: see text] be a commutative ring with identity. The prime ideal sum graph of [Formul... more Let [Formula: see text] be a commutative ring with identity. The prime ideal sum graph of [Formula: see text], denoted by [Formula: see text], is a graph whose vertices are nonzero proper ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is a prime ideal of [Formula: see text]. In this paper, we study some connections between the graph-theoretic properties of this graph and some algebraic properties of rings. The clique number, the chromatic number and the domination number of the prime ideal sum graph for some classes of rings are studied. It is observed that under which condition [Formula: see text] is complete. Moreover, the diameter and the girth of [Formula: see text] are studied.
From its inception, public-key cryptosystems have been an area of active research. Various aspect... more From its inception, public-key cryptosystems have been an area of active research. Various aspects of public-key encryption like constructions, security notions, adversarial models, hardness assumptions, proof-methodology, efficiency, compatibility etc. have been analysed and re-analysed in the last three and half decades by numerous cryptographers. Some of them are good enough to survive while some of them, though broken, provides meaningful insights towards the subject. In this article, our aim is to provide an expository as well as technical (as far as possible, keeping in mind its brevity) overview of the subject as it has progressed over the years, along with some open problems and suitable references.
Electronic Journal of Graph Theory and Applications
In this paper, we introduce a new family of cubic graphs Γ(m), called Generalized Pappus graphs, ... more In this paper, we introduce a new family of cubic graphs Γ(m), called Generalized Pappus graphs, where m ≥ 3. We compute the automorphism group of Γ(m) and characterize when it is a Cayley graph.
Discrete Mathematics & Theoretical Computer Science, Mar 30, 2022
The determining number of a graph G = (V, E) is the minimum cardinality of a set S ⊆ V such that ... more The determining number of a graph G = (V, E) is the minimum cardinality of a set S ⊆ V such that pointwise stabilizer of S under the action of Aut(G) is trivial. In this paper, we provide some improved upper and lower bounds on the determining number of Kneser graphs. Moreover, we provide the exact value of the determining number for some subfamilies of Kneser graphs.
In this paper, we introduce a new graph, called subgroup product graph SP (G) of a group G. The s... more In this paper, we introduce a new graph, called subgroup product graph SP (G) of a group G. The set of vertices of SP (G) is the set of all non-trivial proper subgroups of G and two vertices H and K are adjacent if and only if HK = G. We characterize when SP (G) is connected, a complete graph, star graph, has an universal vertex etc. We also find various graph parameters like diameter, girth, bipartiteness etc. in terms of properties of G.
Rose window graphs are a family of tetravalent graphs, introduced by Steve Wilson. Following it, ... more Rose window graphs are a family of tetravalent graphs, introduced by Steve Wilson. Following it, Kovacs, Kutnar and Marusic classified the edge-transitive rose window graphs and Dobson, Kovacs and Miklavic characterized the vertex transitive rose window graphs. In this paper, we classify the Cayley rose window graphs.
The determining number of a graph $G = (V,E)$ is the minimum cardinality of a set $S\subseteq V$ ... more The determining number of a graph $G = (V,E)$ is the minimum cardinality of a set $S\subseteq V$ such that pointwise stabilizer of $S$ under the action of $Aut(G)$ is trivial. In this paper, we prove some improved upper and lower bounds on the determining number of Kneser graphs. Moreover, we compute the exact value of the determining number for some subfamilies of Kneser graphs. Finally, we show that the number of Kneser graphs with a given determining number $r$ is an increasing function of $r$.
Discrete Mathematics, Algorithms and Applications, 2021
The neighborhood graph [Formula: see text] of a graph [Formula: see text] has the same vertex set... more The neighborhood graph [Formula: see text] of a graph [Formula: see text] has the same vertex set as [Formula: see text] and two vertices are adjacent in [Formula: see text] if and only if they have a common neighbor in [Formula: see text]. We study the domination number, independence number and chromatic number of [Formula: see text] in terms of those of [Formula: see text]. Moreover we investigate various structural similarities between [Formula: see text] and [Formula: see text].
In this paper, we initiate the study of Paley-type graphs \(\Gamma_N\) modulo \(N=pq\), where \(p... more In this paper, we initiate the study of Paley-type graphs \(\Gamma_N\) modulo \(N=pq\), where \(p,q\) are distinct primes of the form \(4k+1\). It is shown that \(\Gamma_N\) is an edge-regular, symmetric, Eulerian and Hamiltonian graph. Also, the vertex connectivity, edge connectivity, diameter and girth of \(\Gamma_N\) are studied and their relationship with the forms of \(p\) and \(q\) are discussed. Moreover, we specify the forms of primes for which \(\Gamma_N\) is triangulated or triangle-free and provide some bounds (exact values in some particular cases) for the order of the automorphism group \(\operatorname{Aut}(\Gamma_N)\) of the graph \(\Gamma_N\), the chromatic number, the independence number, and the domination number of \(\Gamma_N\).
In this paper, we introduce the notion of connected fair domination in graphs. A connected fair d... more In this paper, we introduce the notion of connected fair domination in graphs. A connected fair dominating set in a graph G (or \(\mathsf {CFD}\)-set) is a dominating set S such that \(\langle S \rangle \) is connected in G and all vertices not in S are dominated by the same number of vertices from S, i.e., every two vertices not in S has the same number of neighbours in S. The connected fair domination number of G (\(\mathsf {cfd}(G)\)) is the minimum cardinality of a \(\mathsf {CFD}\)-set in G. Apart from finding \(\mathsf {cfd}(G)\) for some standard graphs G, we proved various bounds on \(\mathsf {cfd}(G)\) in terms of order and some other graph parameters of G.
The annihilating-ideal graph of a commutative ring R with unity is defined as the graph AG(R) wit... more The annihilating-ideal graph of a commutative ring R with unity is defined as the graph AG(R) with the vertex set is the set of all non-zero ideals with non-zero annihilators and two distinct vertices I and J are adjacent if and only if IJ = 0. Nikandish et.al. proved that AG(Zn) is weakly perfect. In this short paper, we characterize n for which AG(Zn) is perfect.
In this paper, we introduce a graph structure, called zero-set intersection graph Γ(C(X)), on the... more In this paper, we introduce a graph structure, called zero-set intersection graph Γ(C(X)), on the ring of real valued continuous functions, C(X), on a Tychonoff space X. We show that the graph is connected and triangulated. We also study the inter-relationship of cliques of Γ(C(X)) and ideals in C(X) which helps to characterize the structure of maximal cliques of Γ(C(X)) by different kind of maximal ideals of C(X). We show that there are at least 2c many different maximal cliques which are never graph isomorphic to each other. Furthermore, we study the neighbourhood properties of a vertex and show its connection with the topology of X and algebraic properties of C(X). Finally, it is shown that two graphs are isomorphic if and only if the corresponding rings are isomorphic if and only if the corresponding topologies are homeomorphic either for first countable topological spaces or for realcompact topological spaces.
The co-maximal subgroup graph Γ(G) of a group G is a graph whose vertices are nontrivial proper s... more The co-maximal subgroup graph Γ(G) of a group G is a graph whose vertices are nontrivial proper subgroups of G and two vertices H and K are adjacent if HK = G. In this paper, we continue the study of Γ(G), especially when Γ(G) has isolated vertices. We define a new graph Γ∗(G), which is obtained by removing isolated vertices from Γ(G). We characterize when Γ∗(G) is connected, a complete graph, star graph, has an universal vertex etc. We also find various graph parameters like diameter, girth, bipartiteness etc. in terms of properties of G.
In this paper, we introduce a new family of graphs, $\Gamma(n,a)$. We show that it is an infinite... more In this paper, we introduce a new family of graphs, $\Gamma(n,a)$. We show that it is an infinite family of tetravalent half-transitive Cayley graphs. Apart from that, we determine some structural properties of $\Gamma(n,a)$.
In this short paper, we characterize the positive integers n for which intersection graph of idea... more In this short paper, we characterize the positive integers n for which intersection graph of ideals of ℤn is perfect.
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