AKCE International journal of Graphs and Combinatorics, 2024
The concept of the covering energy of a poset is known. In this paper, we obtain a relation betwe... more The concept of the covering energy of a poset is known. In this paper, we obtain a relation between the characteristic polynomials of the linear sum of two bounded posets and its components. Also, relations between covering energies of given posets and their linear sum as well as vertical sum are obtained. Eigenvalues and covering energies of some particular types of posets are also studied.
Journal of the Indian Mathematical Society, Mar 24, 2023
In this note we provide a solution to the problem “Find a structural characterization of cubic gr... more In this note we provide a solution to the problem “Find a structural characterization of cubic graph for which the isolate domination number equals one plus its domination number.” We show that if G is a cubic graph of order n and if 6 | n, then the isolate domination number of G is the same as the domination number of G. We also prove that if G is a connected cubic graph with diam(G) > 2, then the isolate domination number is the same as the domination number.
Asian-european Journal of Mathematics, Nov 7, 2019
We introduce the concepts of a hyper chromatic, a critically chromatic and a purely chromatic lat... more We introduce the concepts of a hyper chromatic, a critically chromatic and a purely chromatic lattice. We obtain some characterizations for purely chromatic and hyper chromatic dismantlable lattices. We prove relationships between the chromatic number of two lattices and their linear sum, vertical sum and adjunct.
Journal of the Indian Mathematical Society, Dec 1, 2016
A complete classification of the class of dismantlable lattices in terms of chromatic numbers is ... more A complete classification of the class of dismantlable lattices in terms of chromatic numbers is given. In fact, it is proved that a dismantlable lattice is at most 3-chromatic and the class of 2-Chromatic dismantlable lattices is characterized by using the structure theorem for dismantlable lattices.
Journal of the Indian Mathematical Society, Jul 1, 2020
The concept of the covering energy of a poset is introduced and its bounds are given. We compute ... more The concept of the covering energy of a poset is introduced and its bounds are given. We compute covering energy of some classes of posets like Sn, 2n. The posets Dk and D'k are defined and two recurrence relations for the characteristic polynomials of these posets are obtained. The energies of the posets D1, D2, D3, D4 and D5 are explicitly computed. The existence of some eigenvalues for some type of Dk and D'k is proved.
In this note we provide a solution to the problem “Find a structural characterization of cubic gr... more In this note we provide a solution to the problem “Find a structural characterization of cubic graph for which the isolate domination number equals one plus its domination number.” We show that if G is a cubic graph of order n and if 6 | n, then the isolate domination number of G is the same as the domination number of G. We also prove that if G is a connected cubic graph with diam(G) > 2, then the isolate domination number is the same as the domination number.
We introduce the concepts of a hyper chromatic, a critically chromatic and a purely chromatic lat... more We introduce the concepts of a hyper chromatic, a critically chromatic and a purely chromatic lattice. We obtain some characterizations for purely chromatic and hyper chromatic dismantlable lattices. We prove relationships between the chromatic number of two lattices and their linear sum, vertical sum and adjunct.
A complete classification of the class of dismantlable lattices in terms of chromatic numbers is ... more A complete classification of the class of dismantlable lattices in terms of chromatic numbers is given. In fact, it is proved that a dismantlable lattice is at most 3-chromatic and the class of 2-Chromatic dismantlable lattices is characterized by using the structure theorem for dismantlable lattices.
The aim of this paper is to introduce the concept of minimum independent dominating energy of a g... more The aim of this paper is to introduce the concept of minimum independent dominating energy of a graph, compute it for some classes of graphs and to study its bounds.
The Journal of the Indian Mathematical Society, 2020
The concept of the covering energy of a poset is introduced and its bounds are given. We compute ... more The concept of the covering energy of a poset is introduced and its bounds are given. We compute covering energy of some classes of posets like Sn, 2n. The posets Dk and D'k are defined and two recurrence relations for the characteristic polynomials of these posets are obtained. The energies of the posets D1, D2, D3, D4 and D5 are explicitly computed. The existence of some eigenvalues for some type of Dk and D'k is proved.
We introduce the concepts of a hyper chromatic, a critically chromatic and a purely chromatic lat... more We introduce the concepts of a hyper chromatic, a critically chromatic and a purely chromatic lattice. We obtain some characterizations for purely chromatic and hyper chromatic dismantlable lattices. We prove relationships between the chromatic number of two lattices and their linear sum, vertical sum and adjunct.
The concept of `adjunct' operation of two lattices with respect to a pair of elem... more The concept of `adjunct' operation of two lattices with respect to a pair of elements is introduced. A structure theorem namely, `A finite lattice is dismantlable if and only if it is an adjunct of chains' is obtained. Further it is established that for any adjunct representation of a dismantlable lattice the number of chains as well as the number
The concept of coloring is studied for graphs derived from lattices with 0. It is shown that, if ... more The concept of coloring is studied for graphs derived from lattices with 0. It is shown that, if such a graph is derived from an atomic or distributive lattice, then the chromatic number equals the clique number. If this number is finite, then in the case of a distributive lattice, it is determined by the number of minimal prime ideals in the lattice. An estimate for the number of edges in such a graph of a finite lattice is given.
AKCE International journal of Graphs and Combinatorics, 2024
The concept of the covering energy of a poset is known. In this paper, we obtain a relation betwe... more The concept of the covering energy of a poset is known. In this paper, we obtain a relation between the characteristic polynomials of the linear sum of two bounded posets and its components. Also, relations between covering energies of given posets and their linear sum as well as vertical sum are obtained. Eigenvalues and covering energies of some particular types of posets are also studied.
Journal of the Indian Mathematical Society, Mar 24, 2023
In this note we provide a solution to the problem “Find a structural characterization of cubic gr... more In this note we provide a solution to the problem “Find a structural characterization of cubic graph for which the isolate domination number equals one plus its domination number.” We show that if G is a cubic graph of order n and if 6 | n, then the isolate domination number of G is the same as the domination number of G. We also prove that if G is a connected cubic graph with diam(G) > 2, then the isolate domination number is the same as the domination number.
Asian-european Journal of Mathematics, Nov 7, 2019
We introduce the concepts of a hyper chromatic, a critically chromatic and a purely chromatic lat... more We introduce the concepts of a hyper chromatic, a critically chromatic and a purely chromatic lattice. We obtain some characterizations for purely chromatic and hyper chromatic dismantlable lattices. We prove relationships between the chromatic number of two lattices and their linear sum, vertical sum and adjunct.
Journal of the Indian Mathematical Society, Dec 1, 2016
A complete classification of the class of dismantlable lattices in terms of chromatic numbers is ... more A complete classification of the class of dismantlable lattices in terms of chromatic numbers is given. In fact, it is proved that a dismantlable lattice is at most 3-chromatic and the class of 2-Chromatic dismantlable lattices is characterized by using the structure theorem for dismantlable lattices.
Journal of the Indian Mathematical Society, Jul 1, 2020
The concept of the covering energy of a poset is introduced and its bounds are given. We compute ... more The concept of the covering energy of a poset is introduced and its bounds are given. We compute covering energy of some classes of posets like Sn, 2n. The posets Dk and D'k are defined and two recurrence relations for the characteristic polynomials of these posets are obtained. The energies of the posets D1, D2, D3, D4 and D5 are explicitly computed. The existence of some eigenvalues for some type of Dk and D'k is proved.
In this note we provide a solution to the problem “Find a structural characterization of cubic gr... more In this note we provide a solution to the problem “Find a structural characterization of cubic graph for which the isolate domination number equals one plus its domination number.” We show that if G is a cubic graph of order n and if 6 | n, then the isolate domination number of G is the same as the domination number of G. We also prove that if G is a connected cubic graph with diam(G) > 2, then the isolate domination number is the same as the domination number.
We introduce the concepts of a hyper chromatic, a critically chromatic and a purely chromatic lat... more We introduce the concepts of a hyper chromatic, a critically chromatic and a purely chromatic lattice. We obtain some characterizations for purely chromatic and hyper chromatic dismantlable lattices. We prove relationships between the chromatic number of two lattices and their linear sum, vertical sum and adjunct.
A complete classification of the class of dismantlable lattices in terms of chromatic numbers is ... more A complete classification of the class of dismantlable lattices in terms of chromatic numbers is given. In fact, it is proved that a dismantlable lattice is at most 3-chromatic and the class of 2-Chromatic dismantlable lattices is characterized by using the structure theorem for dismantlable lattices.
The aim of this paper is to introduce the concept of minimum independent dominating energy of a g... more The aim of this paper is to introduce the concept of minimum independent dominating energy of a graph, compute it for some classes of graphs and to study its bounds.
The Journal of the Indian Mathematical Society, 2020
The concept of the covering energy of a poset is introduced and its bounds are given. We compute ... more The concept of the covering energy of a poset is introduced and its bounds are given. We compute covering energy of some classes of posets like Sn, 2n. The posets Dk and D'k are defined and two recurrence relations for the characteristic polynomials of these posets are obtained. The energies of the posets D1, D2, D3, D4 and D5 are explicitly computed. The existence of some eigenvalues for some type of Dk and D'k is proved.
We introduce the concepts of a hyper chromatic, a critically chromatic and a purely chromatic lat... more We introduce the concepts of a hyper chromatic, a critically chromatic and a purely chromatic lattice. We obtain some characterizations for purely chromatic and hyper chromatic dismantlable lattices. We prove relationships between the chromatic number of two lattices and their linear sum, vertical sum and adjunct.
The concept of `adjunct' operation of two lattices with respect to a pair of elem... more The concept of `adjunct' operation of two lattices with respect to a pair of elements is introduced. A structure theorem namely, `A finite lattice is dismantlable if and only if it is an adjunct of chains' is obtained. Further it is established that for any adjunct representation of a dismantlable lattice the number of chains as well as the number
The concept of coloring is studied for graphs derived from lattices with 0. It is shown that, if ... more The concept of coloring is studied for graphs derived from lattices with 0. It is shown that, if such a graph is derived from an atomic or distributive lattice, then the chromatic number equals the clique number. If this number is finite, then in the case of a distributive lattice, it is determined by the number of minimal prime ideals in the lattice. An estimate for the number of edges in such a graph of a finite lattice is given.
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