5 citations · 20 across the 12 of their papers we have counts for
12 papers
Algebraic invariants of edge ideals of cubic circulant graphs
Bakhtawar Shaukat, Muhammad Ishaq, Ahtsham ul Haq +1
We obtain the exact values for depth and projective dimension and lower bounds for Stanley depth of the quotient rings of the edge ideals associated with all cubic circulant graphs…
Algebraic Properties of Edge Ideals of Corona Product of Certain Graphs
Bakhtawar Shaukat, Muhammad Ishaq, Ahtasham Ul Haq +1
We study the algebraic invariants namely depth, Stanley depth, regularity and projective dimension of the residue class rings of the edge ideals associated with the corona product…
Depth and Stanley depth of powers of the edge ideals of some caterpillar and lobster trees
Tooba Zahid, Zunaira Sajid, Muhammad Ishaq
Let be a ring of polynomials in finitely many variables over a field. In this paper we give lower bounds for depth and Stanley depth of modules of the type for …
Some algebraic invariants of the residue class rings of the edge ideals of perfect semiregular trees
Bakhtawar Shaukat, Ahtsham Ul Haq, Muhammad Ishaq
Let S be a polynomial algebra over a field. If I is the edge ideal of a perfect semiregular tree, then we give precise formulas for values of depth, Stanley depth, projective dimen…
Depth and Stanley depth of the edge ideals of the strong product of some graphs
Zahid Iqbal, Muhammad Ishaq, Muhammad Ahsan Binyamin
In this paper we study depth and Stanley depth of the edge ideals and quotient rings of the edge ideals, associated to classes of graphs obtained by taking the strong product of tw…
PD-ML-Lite: Private Distributed Machine Learning from Lighweight Cryptography
Maksim Tsikhanovich, Malik Magdon-Ismail, Muhammad Ishaq +1
Privacy is a major issue in learning from distributed data. Recently the cryptographic literature has provided several tools for this task. However, these tools either reduce the q…