The classical Turán-type extremal problem asks for the maximum number of edges in an \(F\)-free graph on \(n\) vertices, along with a characterization of the extremal graphs, where \(F\) is a fixed graph (or class of graphs). The spectral version, introduced by Nikiforov, asks for bounds on the maximum spectral radius of an \(F\)-free graph on \(n\) vertices. One can also pose analogous questions for \(F\)-free graphs on \(m\) edges; this is the Brualdi–Hoffman-type variant.

The triangle case (\(F = K_3\)) has received considerable attention in the recent literature. M. Rajesh Kannan and I focus on the case \(F = K_4\), the next case in the complete-graph progression.

The problem of representing a graph as the intersection graph of a collection of geometric or combinatorial objects is well studied — both theoretically and from computational and practical standpoints. Our focus was on representing a given graph optimally (in a precise sense) as the vertex-intersection graph of a collection of paths.

Such representations always exist, but finding optimal ones, or even tight bounds on their order, does not seem to be an easy problem. Several questions in this area remain open. See Balachandran et al. (in preparation).

My doctoral work spanned three threads in extremal and algebraic combinatorics.

Surface coloring. I studied the gap between the chromatic and list chromatic numbers for graphs embeddable on a surface. In particular, I completed the classification of the colorability of the \(6\)-regular triangulations of the torus and gave a linear-time \(5\)-list-coloring algorithm for a large class of these triangulations. See Sankarnarayanan (2022), Balachandran–Sankarnarayanan (2021), and Balachandran–Sankarnarayanan (2026).

Extremal set theory. I studied a fractional variant of \(L\)-intersecting set systems and made progress on tight upper bounds for the sizes of such families. See Balachandran et al. (2023b) and Balachandran et al. (2025).

Combinatorial matrix theory. I studied the ranks of a class of symmetric, zero-diagonal matrices that arise from tournaments on \([n]\). These matrices have rank at least linear in \(n\) with high probability, and I characterized their rank in several extremal cases. The constructions also yield bisection-closed families that improve previous bounds, connecting back to extremal set theory. See Balachandran et al. (2023a), Balachandran et al. (2024), and Balachandran–Sankarnarayanan (2024).

Full thesis: Some problems in combinatorics: Excursions in graph colorings and extremal set theory.

Prior to my PhD, I briefly worked as a project assistant with Kumar Balasubramanian at IISER Bhopal, studying a small amount of \(p\)-adic representation theory; see Balasubramanian et al. (2019). My Master's thesis, also at IISER Bhopal, was on automorphic forms and Tate's thesis, guided by Karam Deo Shankhadhar. During my undergraduate studies, I also explored some number theory with Brundaban Sahu.