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Cospectral and hyper-energetic self complementary comparability graphs
Merajuddin,S. A. K. Kirmani,Parvez Ali,S. Pirzada 한국산업응용수학회 2007 Journal of the Korean Society for Industrial and A Vol.11 No.3
A graph G is self-complementary (sc) if it is isomorphic to its complement. G is perfect if for all induced subgraphs H of G, the chromatic number of H (denoted X(H)) equals the number of vertices in the largest clique in H (denoted w(H)). An sc graph which is also perfect is known as sc perfect graph. A comparability graph is an undirected graph if it can be oriented into transitive directed graph. An sc comparability (sec) is clearly a subclass of sc perfect graph. In this paper we show that no two non-isomorphic sec graphs with n vertices each, (n < 13) have same spectrum, and that the smallest positive integer for which there exists hyper-energetic sec graph is 13.
Cospectral and hyper-energetic self complementary comparability graphs
( Merajuddin ),( S. A. K. Kirmani ),( Parvez Ali ),( S. Pirzada ) 한국산업응용수학회(구 한국산업정보응용수학회) 2007 한국산업정보응용수학회 Vol.11 No.3
A graph G is self-complementary (sc) if it is isomorphic to its complement. G is perfect if for all induced subgraphs H of G, the chromatic number of H (denoted X(H)) equals the number of vertices in the largest clique in H (denoted w(H)). An sc graph which is also perfect is known as sc perfect graph. A comparability graph is an undirected graph if it can be oriented into transitive directed graph. An sc comparability (scc) is clearly a subclass of sc perfect graph. In this paper we show that no two non-isomorphic scc graphs with n vertices each, (n < 13) have same spectrum, and that the smallest positive integer for which there exists hyper-energetic scc graph is 13.
GENERALIZED DOMINOES TILING'S MARKOV CHAIN MIXES FAST
KAYIBI, K.K.,SAMEE, U.,MERAJUDDIN, MERAJUDDIN,PIRZADA, S. The Korean Society for Computational and Applied M 2019 Journal of applied mathematics & informatics Vol.37 No.5
A generalized tiling is defined as a generalization of the properties of tiling a region of ${\mathbb{Z}}^2$ with dominoes, and comprises tiling with rhombus and any other tilings that admits height functions which can be ordered into a distributive lattice. By using properties of the distributive lattice, we prove that the Markov chain consisting of moving from one height function to the next by a flip is fast mixing and the mixing time ${\tau}({\epsilon})$ is given by ${\tau}({\epsilon}){\leq}(kmn)^3(mn\;{\ln}\;k+{\ln}\;{\epsilon}^{-1})$, where mn is the area of the grid ${\Gamma}$ that is a k-regular polycell. This result generalizes the result of the authors (T-tetromino tiling Markov chain is fast mixing, Theor. Comp. Sci. (2018)) and improves on the mixing time obtained by using coupling arguments by N. Destainville and by M. Luby, D. Randall, A. Sinclair.
ON SIGNLESS LAPLACIAN SPECTRUM OF THE ZERO DIVISOR GRAPHS OF THE RING ℤ<sub>n</sub>
Pirzada, S.,Rather, Bilal A.,Shaban, Rezwan Ul,Merajuddin, Merajuddin The Kangwon-Kyungki Mathematical Society 2021 한국수학논문집 Vol.29 No.1
For a finite commutative ring R with identity 1 ≠ 0, the zero divisor graph (R) is a simple connected graph having vertex set as the set of nonzero zero divisors of R, where two vertices x and y are adjacent if and only if xy = 0. We find the signless Laplacian spectrum of the zero divisor graphs (ℤn) for various values of n. Also, we find signless Laplacian spectrum of (ℤn) for n = pz, z ≥ 2, in terms of signless Laplacian spectrum of its components and zeros of the characteristic polynomial of an auxiliary matrix. Further, we characterise n for which zero divisor graph (ℤn) are signless Laplacian integral.
GENERALIZED DOMINOES TILING'S MARKOV CHAIN MIXES FAST
K.K. Kayibi,U. Samee,Merajuddin,S. PIRZADA 한국전산응용수학회 2019 Journal of applied mathematics & informatics Vol.37 No.5
A generalized tiling is defined as a generalization of the properties of tiling a region of Z^2 with dominoes, and comprises tiling with rhombus and any other tilings that admits height functions which can be ordered into a distributive lattice. By using properties of the distributive lattice, we prove that the Markov chain consisting of moving from one height function to the next by a flip is fast mixing and the mixing time τ(ε) is given by τ(ε)≤(kmn)^3 (mn ln k + ln ε^-1 ), where mn is the area of the grid Γ that is a k-regular polycell. This result generalizes the result of the authors (T-tetromino tiling Markov chain is fast mixing, Theor. Comp. Sci. (2018)) and improves on the mixing time obtained by using coupling arguments by N. Destainville and by M. Luby, D. Randall, A. Sinclair.