IIT Bhilai

Kastelyn has proved the existence of Pfaffian orientation on any nice (planar) graph, which can also be seen as Pfaffian orientation on a polygonally cellulated punctured sphere (a sphere with one maximal face removed). However, the number of such orientations is not yet known. In this paper, we introduce the notion of Pfaffian orientations on (punctured) polygonally cellulated surfaces, in general, and provide an expression for the number of such orientations. As a corollary, we derive the number of Pfaffian orientations on a polygonally cellulated (punctured) sphere.

A convexity space over a set X is a collection of subsets of X that contains the empty set and is closed under intersections. We investigate Δ-convexity spaces over graphs, where a subset S of V(G) is Δ-convex if it is closed under formation of triangles. The Carathéodory and exchange numbers measure the dimensionality of convexity spaces, and are parameters that are generally hard to compute. We give tight bounds on the Carathéodory and exchange numbers for Δ-convexity spaces in terms of the number of triangles in G, and we also attempt to characterize the extremal graphs.

In a finite lattice \( L \), two elements are said to be complements if their meet is the minimal element \( \hat{0} \) of \( L \) and their join is the maximal element \( \hat{1} \) of \( L \). For \( m \in L \), let \( C(m) \) denote the set of all complements in \( L \). It is known that if \( L \) is non-complemented, i.e., there exists an element in \( L \) that does not have a complement, then the poset \( \bar{L} = (\hat{0}, \hat{1}) \) is contractible. One of our main results shows that if \( \widetilde{H}_k(\bar{L}, K) \neq 0 \) for some \( k > 0 \) and some field \( K \), then for every \( m \in \bar{L} \), there exists an \( m' \in C(m) \), and an \( n \in (\hat{0}, m] \cap C(m') \) such that \( m' \) and \( n \) have “complementary homologies”.

We give a new version of Bj\"{o}rner and Walker's lattice complementation formula. We apply the above result to derive consequences for the Betti numbers of monomial ideals. In addition, we introduce a notion of complementary faces in a simplicial complex and prove that non-complemented simplicial complexes are acyclic.

This is an ongoing joint project with Sara Faridi and Volkmar Welker.

The Johnson graph \( J(n,k) \) is a classical example of how combinatorial structure, linear algebra, and representations of the symmetric group interact through spectral theory. The adjacency operator of \( J(n,k) \) commutes with the action of the symmetric group \( S_n \) on the \( k \)-subsets of \( \{1,2,\dots,n\} \). As a consequence, its spectrum reflects the decomposition of the corresponding permutation representation into irreducible components.

In this talk, we introduce a two-color analogue of the Johnson graph arising from the action of \( S_n \) on exterior powers \( \wedge^k(\mathbb{C}^n) \), viewed combinatorially as a cocycle-twisted action on \( k \)-subsets. We prove that the associated adjacency operator generates the full commutant of this twisted action. We determine its spectrum and show that the eigenvalues explicitly determine the structure constants of the commutant algebra.

Let \( \mathfrak{g} \) be a symmetrizable Kac-Moody Lie algebra over an algebraically closed field \( \mathbb{F} \) of characteristic 0. Kostant Kumar modules are cyclic submodules of the form \( U(\mathfrak{g})(v_{\lambda} \otimes v_{w\mu}) \) of the tensor product \( V(\lambda) \otimes V(\mu) \) of two highest weight irreducible modules over \( \mathfrak{g} \).

We study the module-theoretic aspects of Kostant Kumar modules. There are path-theoretic characterisations of multiplicity of irreducibles occurring in decompositions of the tensor product and Kostant-Kumar modules due to Littelmann and Joseph. An analogous characterisation at the module level is well known for the tensor product in the finite dimensional case. We generalise this to Kostant-Kumar modules. We then give a generators and relations description of Kostant Kumar modules in the finite dimensional and symmetric Kac-Moody algebra case. As an application of these results, we obtain Schur-positivity type results for symmetric functions corresponding to Kostant-Kumar modules.

In this talk, we present a combinatorial interpretation of a distribution of \( (n-1)^{n-1} \), in the context of uprooted spanning trees of the complete graph \( K_n \), which was previously obtained by Chauve--Dulucq--Guibert. We then establish a combinatorial explanation for the distribution of \( m^{n-1} n^{m-1} \), related to spanning trees of the complete bipartite graph \( K_{m,n} \), which seems new. Furthermore, we extend this study to the graph \( K_n \setminus \{e_{1,n}\} \), obtained by deleting an edge from \( K_n \), and derive a new identity for the number of its uprooted spanning trees.

A recent result of Lofano and Paolini expresses the characteristic polynomial of a real hyperplane arrangement in terms of a projection statistic on the regions of the arrangement. We use this result to give an alternative proof for Greene and Zaslavsky's interpretation for the coefficients of the chromatic polynomial of a graph and further generalize this interpretation to signed graphs. We also show that this projection statistic has a nice combinatorial interpretation in the case of the braid arrangement, which generalizes to graphical arrangements of natural unit interval graphs.

A connected graph is matching covered if each edge lies in some perfect matching. There is extensive literature on this class; one may refer to the recent monograph 'Perfect Matchings: A Theory of Matching Covered Graphs' by Lucchesi and Murty (2024). One of the fundamental pillars of this subject is the tight cut decomposition theory. In particular, Lovász (JCT-B, 1987) established that every matching covered graph \( G \) admits a unique decomposition into special ones called bricks (nonbipartite) and braces (bipartite); we use \( b(G) \) to denote its number of bricks. We remark that \( G \) is bipartite if and only if \( b(G) = 0 \).

Certain containment notions such as minors and topological minors play a key role in graph theory. Likewise, there are containment notions that preserve the property of being matching covered. One of them, 'conformal minors', is intrinsically related to the ear decomposition theory of matching covered graphs; Lovász and Plummer showed that every such graph, except \( K_2 \) and cycles, contains either \( \theta \) or \( K_4 \) as a conformal minor, where \( \theta \) is the \( 3 \)-regular graph on two vertices. Moreover, Lovász proved that every nonbipartite matching covered graph (or equivalently, those with \( b \geq 1 \)) contains either \( K_4 \) or triangular prism \( \overline{C_6} \) as a conformal minor. It is worth noting that, in the case of nonbipartite matching covered graphs, Lovász's result implies the aforementioned result of Lovász and Plummer; in this sense, the former may be viewed as a refinement of the latter that is applicable to a smaller class of graphs.

In the same spirit, we present a further refinement of Lovász's result that is applicable to those matching covered graphs that satisfy \( b \geq 2 \). For technical reasons, it is imperative to use a weaker notion of containment, 'matching minors'. In this article, we prove that every such graph contains one of \( 16 \) graphs as a matching minor; each of these \( 16 \) graphs has the (desired) properties that \( b = 2 \) whereas every proper matching minor satisfies \( b \leq 1 \). As a corollary of our main result, we establish Dalwadi's conjecture (2024), which states that every matching covered graph with \( b \geq 2 \) contains a bisubdivision of \( K_{2,3} \) as a subgraph.

The Stanley-Reisner correspondence provides a bridge between combinatorial topology and combinatorial commutative algebra by relating square-free monomial ideals to simplicial complexes. A classical result illustrating this connection is Fröberg’s theorem, which characterizes all square-free monomial ideals generated in degree 2—corresponding to edge ideals of graphs—that have a linear resolution, via the graph being chordal. This naturally raises the question of extending the result to higher-degree square-free monomial ideals. Several attempts have explored this by generalizing the notion of chordal graphs to hypergraphs.

In this talk, I will present this approach and prove Fröberg’s theorem for a certain class of hypergraphs by showing that the associated simplicial complexes of their edge ideals are vertex decomposable. This is a joint work with P. Deshpande and A. Roy.

Let \( G \) be a finite simple graph and consider its edge ideal. The squarefree powers of this ideal are generated by collections of pairwise disjoint edges, that is, by matchings of fixed size. The associated simplicial complex, called the matching-free complex, consists of all vertex subsets that do not contain a matching of a given size.

In this talk, we study these complexes for whisker graphs, obtained by attaching a leaf to every vertex of a base graph. We first give a complete characterization of when the matching-free complex is pure, showing that this happens precisely when the base graph has no short odd cycles. We then determine when the complex is shellable in terms of the girth of the base graph.

As consequences, we obtain new results on Cohen-Macaulayness and sequential Cohen-Macaulayness of squarefree powers of edge ideals of whisker graphs. Finally, we verify a recent conjecture on the depth of squarefree powers for whisker cycles in the expected range and extend the result to all whisker graphs.

For a metric space \( X \) and \( r \geq 0 \), the Vietoris-Rips complex \( \mathcal{VR}(X;r) \) is a simplicial complex whose simplices are finite subsets of \( X \) with diameter at most \( r \). Vietoris-Rips complexes have applications in various places, including data analysis, geometric group theory, sensor networks, etc. Consider the integer lattice \( \mathbb{Z}^n \) as a metric space equipped with the \( d_1 \)-metric (standard word metric in the Cayley graph). Recently, Matthew Zaremsky conjectured that \( \mathcal{VR}(\mathbb{Z}^n;r) \) is contractible for \( r \geq n \).

In this talk, we discuss recent progress on Zaremsky’s conjecture. In particular, we show that the conjecture holds for all \( n \leq 5 \). Moreover, we determine the homotopy type of \( \mathcal{VR}(\mathbb{Z}^n;2) \) and show that these complexes are homotopy equivalent to a wedge of countably infinite copies of 3-spheres \( S^3 \).

This is a joint work with Raju Kumar Gupta and Sourav Sarkar.

We introduce marked multi-colorings, marked chromatic polynomials, and marked (multivariate) independence series for hypergraphs, showing that the coefficients of the \( q \)-th power of the marked independence series recover the marked chromatic polynomial, thereby generalizing a known result for graphs to hypergraphs.

We extend these constructions to subspace arrangements, prove that the number of marked multi \( q \)-colorings is polynomial in \( q \), and establish non-negativity of coefficients in the \( (-q) \)-th power of the independence series for hyperplane arrangements.

The restriction problem in algebraic combinatorics seeks a combinatorial interpretation of the coefficients that arise when an irreducible polynomial representation of \( GL_{n}(\mathbb{C}) \) (equivalently a module for the Schur algebra \( S(n,d) \)) is restricted to the symmetric group. In 1940, Littlewood gave a plethystic formula describing these restriction coefficients. In this talk, we investigate an analogous problem in the super setting. More precisely, we consider the restriction of irreducible modules of the Schur superalgebra \( S(m,n;d) \) to representations of \( S_{m} \times S_{n} \). We present a plethystic formula for the corresponding restriction coefficients, thereby extending Littlewood’s classical result to the superalgebra framework.

A connected graph is matching covered (aka \( 1 \)-extendable) if each edge extends to (that is, belongs to) some perfect matching. For instance, Schönberger (1934) proved that every \( 2 \)-connected cubic graph is matching covered. There is extensive literature on this class; one may refer to 'Perfect Matchings: A Theory of Matching Covered Graphs' by Lucchesi and Murty (2024).

A subgraph \( H \) of a graph \( G \) is conformal if \( G - V(H) \) has a perfect matching. Conformal subgraphs arise naturally from the ear decomposition theory due to Hetyei, Lovász and Plummer, and play an important role in the subject; for example, Pfaffian orientations may be defined solely in terms of conformal cycles. A matching covered graph is cycle-extendable if the following equivalent conditions hold: (1) each even cycle is conformal, (2) each perfect matching of any even cycle extends to a perfect matching, (3) each even cycle may be written as the symmetric difference of two perfect matchings, (4) every even cycle extends to an 'ear decomposition'.

Despite the strict restriction, no \( \textbf{NP} \)-characterization of cycle-extendable graphs is known. However, the problem has been solved for certain graph classes; in particular, the claw-free case by Zhang, Wang and Yuan (DMTCS 2020), and the planar case by Dalwadi, Pause, Diwan and Kothari (DMTCS 2025). In a joint work with the late Ajit A. Diwan and Nishad Kothari, we solved the problem for the bipartite cubic case; we are now working towards settling the entire cubic case.

A \( 2 \)-connected cubic graph is essentially \( 4 \)-edge-connected if its only \( 3 \)-cuts are the trivial ones (that is, all edges incident at a vertex); when restricted to bipartite graphs, these are precisely the cubic braces. We show that in order to characterize all bipartite cubic cycle-extendable graphs, it suffices to characterize just the cycle-extendable cubic braces. We prove that for each vertex \( v \) of any cubic brace \( G \), there exists a cycle \( Q \) that avoids \( v \) but contains each of its neighbours, unless \( G \) is either \( K_{3,3} \), \( \mathbb{C}_4 \) (the unique cubic graph whose underlying simple graph is \( C_4 \)), or \( \theta \) (the unique cubic graph whose underlying simple graph is \( K_2 \)); clearly, such a cycle \( Q \) is not conformal. As a consequence, a bipartite cubic graph, except \( \theta \), is cycle-extendable if and only if it may be obtained from \( K_{3,3} \) and \( \mathbb{C}_4 \) by repeated application of the 'splicing' operation.

A crystallization of a PL \( n \)-manifold (possibly with \( h \) boundary components) is an \( (n+1) \)-colored graph encoding that manifold. Classifying which \( (n+1) \)-colored graphs represent PL \( n \)-manifolds is a fundamental problem in combinatorial topology. For \( n=3 \), there are well-established combinatorial criteria (including versions for manifolds with boundary), whereas for \( n \ge 4 \), no general characterization is known and the problem remains open.

In this presentation, we introduce a new algorithm to compute the fundamental group of a \( 3 \)-manifold (possibly with connected boundary) directly from its crystallization. By combining this algorithmic criterion with the classical fact that a \( 3 \)-manifold with trivial fundamental group is the \( 3 \)-sphere (or the \( 3 \)-ball when the boundary is connected), we obtain a combinatorial characterization, suitable for practical verification, of when a \( 5 \)-colored graph is a crystallization of a \( 4 \)-manifold (possibly with \( h \) boundary components). Our characterization provides an effective method to determine whether a given \( 5 \)-colored graph represents a \( 4 \)-manifold.

The chessboard complex \( \Delta_{m,n} \) is a simplicial complex arising from matchings in the complete bipartite graph with parts of size \( m \) and \( n \). In this talk, we study the symbolic powers of the cover ideal associated with \( \Delta_{m,n} \). We provide a complete characterization, in terms of the parameters \( m \) and \( n \), describing when all symbolic powers of the cover ideal are componentwise linear.

The degree of a map between orientable manifolds is a fundamental concept in topology, providing deep insights into the structure of manifolds and the behavior of maps between them. Recently, this notion has been extensively studied, particularly in the context of simplicial maps between orientable triangulable spaces.

In this poster, we focus on the construction of non-degenerate simplicial maps of degree \( d \in \mathbb{Z} \) on \( n \)-spheres for \( n \geq 2 \). We develop a general method, based on connected sums and facet orientations, to construct simplicial maps of any prescribed degree \( d \in \mathbb{Z} \) between triangulated spheres.

We investigate the asymptotic behavior of \( \Lambda(n,d) \), defined as the minimum number of vertices required for a triangulated \( n \)-sphere to admit a simplicial map of degree \( d \) to \( \mathbb{S}^n_{n+2} \), for \( n \geq 3 \) and \( d \geq 1 \). As a consequence, we answer a question posed by Ryabichev in his paper. In addition to vertex-minimal constructions, we obtain facet-minimal degree maps for large degrees. Specifically, for each \( d \geq n^2 + 1 \), we construct a simplicial map of degree \( d \) from a triangulated \( n \)-sphere with \( d(n+2) \) facets to \( \mathbb{S}^n_{n+2} \), for \( n \geq 3 \).

As an application of the constructions, we derive improved bounds on the covering type of Moore spaces. Finally, we conclude with several open questions that may be of independent interest.

Let \( G \) be a finite non-cyclic group. Define \( \mathrm{Cyc}(G) \) as the set of all elements \( a \in G \) such that for any \( b \in G \), the subgroup \( \langle a, b \rangle \) is cyclic. The non-cyclic graph \( \Gamma(G) \) of \( G \) is a simple undirected graph with vertex set \( G \setminus \mathrm{Cyc}(G) \), where two distinct vertices \( x \) and \( y \) are adjacent if the subgroup \( \langle x, y \rangle \) is not cyclic.

An independent subset \( C \) of the vertex set of a graph \( \Gamma \) is called a perfect code of \( \Gamma \) if every vertex of \( V(\Gamma)\setminus C \) is adjacent to exactly one vertex in \( C \). A subset \( T \) of the vertex set of a graph \( \Gamma \) is said to be a total perfect code if every vertex of \( \Gamma \) is adjacent to exactly one vertex in \( T \).

In this paper, we prove that the graph \( \Gamma(G) \) is Hamiltonian for any finite non-cyclic nilpotent group \( G \). Also, we characterize all finite groups such that their non-cyclic graphs admit a perfect code. Finally, we prove that for a non-cyclic nilpotent group \( G \), the non-cyclic graph \( \Gamma(G) \) does not admit a total perfect code.

The power graph of a finite group \( G \) is a simple undirected graph with vertex set \( G \), where two vertices are adjacent if one is a power of the other. The intersection power graph of a finite group \( G \) is a simple undirected graph with vertex set \( G \), where two vertices \( x, y \) are adjacent if \( \langle x \rangle \cap \langle y \rangle \neq \{e\} \).

The difference graph \( D(G) \) of a finite group \( G \) is the difference of the intersection power graph \( G_1(G) \) and power graph \( P(G) \), with all isolated vertices removed. We characterize all the finite nilpotent groups \( G \) such that the difference graph is planar. Further, we determine all the finite nilpotent groups whose difference graph has genus at most 2.

Moreover, we prove that there does not exist any group whose difference graph has cross-cap one.

Let \( I \) be any square-free monomial ideal, and \( \mathcal{H}_I \) denote the hypergraph associated with \( I \). Refining the concept of \( k \)-admissible matching of a graph defined by Erey and Hibi, we introduce the notion of generalized \( k \)-admissible matching for any hypergraph.

Using this, we provide a sharp lower bound on the Castelnuovo-Mumford regularity of \( I^{[k]} \), where \( I^{[k]} \) denotes the \( k^{\text{th}} \) square-free power of \( I \). In the special case when \( I \) is equigenerated in degree \( d \), this lower bound can be described using a combinatorial invariant \( \mathrm{aim}(\mathcal{H}_I,k) \).

Specifically, we prove that \( \mathrm{reg}(I^{[k]}) \ge (d-1)\mathrm{aim}(\mathcal{H}_I,k) + k \), whenever \( I^{[k]} \) is non-zero. We further discuss cases including forests, block graphs, and Cohen-Macaulay chordal graphs, where this bound is attained.

The difference subgroup graph \( D(G) \) of a finite group \( G \) is defined as the graph whose vertices are the non-trivial proper subgroups of \( G \), with two distinct vertices \( H \) and \( K \) adjacent if and only if \( \langle H, K \rangle = G \) but \( HK \ne G \). This graph arises naturally as the difference between the join graph \( \Delta(G) \) and the comaximal subgroup graph \( \Gamma(G) \). In this paper, we initiate a systematic study of \( D(G) \) and its reduced version \( D^*(G) \), obtained by removing isolated vertices.

We establish several fundamental structural properties of these graphs, including conditions for connectivity, forbidden subgraph characterizations, and relationships between graph parameters—such as independence number, clique number, and girth—and the solvability or nilpotency of the underlying group.

The paper concludes with a discussion of open problems and potential directions for future research.

Face-number invariants are fundamental in combinatorial topology, with the invariant \( g_2 \) playing a central role. For a normal \( d \)-pseudomanifold \( K \) (\( d \ge 3 \)), it is known that \( g_2(K) \ge g_2(\mathrm{lk}(v,K)) \) for every vertex \( v \). Moreover, if \( K \) has at most two singular vertices and satisfies \( g_2(K)=g_2(\mathrm{lk}(t,K)) \) at a singular vertex \( t \), then \( g_3(K) \ge g_3(\mathrm{lk}(t,K)) \). We call \( K \) \( g_2 \)- and \( g_3 \)-optimal if \( g_2(K)=g_2(\mathrm{lk}(t,K)) \) and \( g_3(K)=g_3(\mathrm{lk}(t,K)) \), for some singular vertex \( t \).

In this poster, we establish structural results for normal \( 4 \)-pseudomanifolds under \( g_2 \)- and \( g_3 \)-optimality conditions. We prove that if \( K \) is a normal \( 4 \)-pseudomanifold with exactly one singular vertex \( t \) and is \( g_2 \)- and \( g_3 \)-optimal at \( t \), then \( K \) can be obtained from the boundary complexes of \( 5 \)-simplices via a sequence of vertex foldings and connected sum operations.

If \( K \) has exactly two singular vertices and is \( g_2 \)- and \( g_3 \)-optimal at one of them, then \( K \) can be derived from the boundary complexes of \( 4 \)-simplices through a sequence of one-vertex suspensions, vertex foldings, and connected sums.

Moreover, in this two-singularity case, we provide an alternative characterization. We prove that such a complex \( K \) can also be obtained from the boundary complexes of \( 5 \)-simplices via a sequence of vertex foldings, edge foldings, and connected sums.

The difference graph \( \mathcal{D}(G) \) of a finite group \( G \) is the difference of the enhanced power graph \( \mathcal{P}_E(G) \) and the power graph \( \mathcal{P}(G) \), with all the isolated vertices removed. In this paper, we characterize the vertex set of the difference graph of finite nilpotent groups and obtain its cardinality. Consequently, we obtain the metric dimension of the difference graph of finite nilpotent groups. Moreover, this paper determines the metric dimension of the difference graphs of certain non-nilpotent groups, namely: dihedral groups, the generalized quaternion groups, and the semi-dihedral groups.

The arboreal gas with fugacity \( \lambda \) is the probability measure on forests in a graph that assigns weight proportional to \( \lambda^{\# \text{edges}} \) to a forest. A longstanding conjecture due to Kahn says that edge events in the arboreal gas are negatively correlated, similar to the case of spanning tree measures (the case when \( \lambda = \infty \)). I will show that the conjecture is true when \( \lambda \) is sufficiently large unless the graph has a very specific structure.

We believe, and this might seem counterintuitive, that our approach provides a strategy to refute the conjecture in full generality. This is joint work with Recep Altar Ciceksiz.

The weakly zero-divisor graph \( W\Gamma(R) \) of a commutative ring \( R \) is the simple undirected graph whose vertices are nonzero zero-divisors of \( R \), and two distinct vertices \( x \) and \( y \) are adjacent if and only if there exist \( w \in \mathrm{ann}(x) \) and \( z \in \mathrm{ann}(y) \) such that \( wz = 0 \). In this manuscript, we determine the vertex connectivity of \( W(\Gamma(R)) \) when: (i) \( R \) is a reduced ring and (ii) \( R \) is an Artinian ring. For such rings \( R \), we also characterize the vertices of minimum degree and the minimum cut-sets of the zero-divisor graph of \( R \).

A diagonally symmetric alternating sign matrix (DSASM) is a symmetric matrix with entries −1, 0 and 1, where the nonzero entries alternate in sign along each row and column, and the sum of the entries in each row and column equals 1. An off-diagonally symmetric alternating sign matrix (OSASM) is a DSASM, where the number of nonzero diagonal entries is 0 for even-order matrices and 1 for odd-order matrices. Kuperberg (Ann. Math., 2002) studied even-order OSASMs and derived a product formula for counting the number of OSASMs of any fixed even order. Recently (arXiv, 2025), we proved a product formula that enumerates odd-order OSASMs. In this talk, I will present product formulas for the number of OSASMs of any fixed order.

Let \( \Gamma \) be an undirected and simple graph. A set \( S \) of vertices in \( \Gamma \) is called a cyclic vertex cutset of \( \Gamma \) if \( \Gamma - S \) is disconnected and has at least two components containing cycles. If \( \Gamma \) has a cyclic vertex cutset, then it is said to be cyclically separable. The cyclic vertex connectivity of \( \Gamma \) is the minimum of cardinalities of the cyclic vertex cutsets of \( \Gamma \). For any finite group \( G \), the order supergraph \( \mathcal{S}(G) \) is the simple and undirected graph whose vertices are elements of \( G \), and two vertices are adjacent if the order of one divides that of the other. In this paper, we characterize all finite nilpotent groups and various non-nilpotent groups whose order super graphs are cyclically separable.

Let \( G \) be a graph on \( n \) vertices. For a given integer \( k \) such that \( 1 \leq k \leq n \), the \( k \)-token graph \( F_k(G) \) of \( G \) is defined as the graph whose vertices are the \( k \)-subsets of the vertex set of \( G \), and two of them are adjacent whenever their symmetric difference is a pair of adjacent vertices in \( G \). It was proved that the algebraic connectivity of \( F_k(G) \) is equal to the algebraic connectivity of \( G \).

In this talk, we characterize all graphs for which the Laplacian spectral radius of \( G \) equals the Laplacian spectral radius of \( F_k(G) \) and identify classes of token graphs that are Laplacian integral.

Given an interval \( I \), \( m_G(I) \) counts the Laplacian eigenvalues of \( G \) (including multiplicity) in \( I \). Let \( \mathcal{U}_{n,g} \) denote the class of unicyclic graphs on \( n \) vertices with girth \( g \). We characterize \( U \in \mathcal{U}_{n,g} \) with \( m_U[0,1)=\gamma(U) \), where \( \gamma(U) \) is the domination number. We prove that \( m_U[0,2) \leq n-\gamma(U) \) and identify certain equality cases. Moreover, if \( U \ncong C_3 \), then \( m_U(n-1,n] \leq 1 \), with equality completely characterized.

For a positive integer \( k \), the total \( k \)-cut complex of a graph \( G \), denoted as \( \Delta_{k}^t(G) \), is the simplicial complex whose facets are \( \sigma \subseteq V(G) \) such that \( |\sigma| = |V(G)|-k \) and the induced subgraph \( G[V(G) \setminus \sigma] \) does not contain any edge. These complexes were introduced by Bayer et al. [Total cut complexes of graphs. Discrete Comput. Geom., 73(2):500–527, 2024] in connection with commutative algebra. In the same paper, they studied the homotopy types of these complexes for various families of graphs, including cycle graphs \( C_n \), squared cycle graphs \( C_n^2 \), and Cartesian products of complete graphs and path graphs \( K_m \square P_2 \) and \( K_2 \square P_n \).

We extend the work of Bayer et al. for these families of graphs. We focus on the complexes \( \Delta_{2}^t(G) \) and determine the homotopy types of these complexes for three classes of graphs: (i) \( p \)-th powers of cycle graphs \( C_n^p \) (ii) Cartesian products of complete graphs and path graphs \( K_m \square P_n \) and (iii) Cartesian products of complete graphs and cycle graphs \( K_m \square C_n \). Using discrete Morse theory, we show that these complexes are homotopy equivalent to wedges of spheres. We also give the number and dimension of spheres appearing in the homotopy type.

Our result on powers of cycle graphs \( C_n^p \) proves a conjecture of Shen et al. about the homotopy type of the complexes \( \Delta_2^t(C_n^p) \).

The talk will cover the computational techniques used and highlight the key results obtained.

Let \( R \) be a commutative ring with unity. The clean graph \( \textnormal{Cl}(R) \) of a ring \( R \) is the simple undirected graph whose vertices are of the form \( (e,u) \), where \( e \) is an idempotent element and \( u \) is a unit of the ring \( R \), and two vertices \( (e,u) \), \( (f,v) \) of \( \textnormal{Cl}(R) \) are adjacent if and only if \( ef = fe = 0 \) or \( uv = vu = 1 \). In this paper, we discuss the connectivity of clean graphs. After ascertaining the minimum degree and corresponding vertices, we determine the vertex connectivity of the clean graph of an arbitrary ring. Moreover, we provide some necessary conditions for an arbitrary graph to be minimally connected. Consequently, we characterize the rings whose clean graphs are minimally (edge) connected. Finally, we prove that the clean graph of a clean ring \( R \) is minimally (edge) connected if and only if \( R \) is a Boolean ring.

Classical results in equivariant topology (e.g., Borsuk--Ulam theorem, \( \mathbb{Z}_p \)-Borsuk--Ulam theorem etc.) have numerous important applications in combinatorics. In this paper, we prove a Hopf--trace type formula, which is a purely combinatorial identity, involving no homology. This theorem provides a unified combinatorial framework for several results in equivariant topology. In fact, we use the Hopf--trace type formula to prove several combinatorial Borsuk--Ulam type theorems, e.g., \( \mathbb{Z}_p \)-Tucker's lemma, combinatorial degree version of \( \mathbb{Z}_p \)-Tucker's lemma, \( \mathbb{Z}_p \)-Borsuk--Ulam theorems etc. Our proofs are purely combinatorial in the sense that they do not involve homology, cohomology or any other notions from continuous topology. The combinatorial degree version of \( \mathbb{Z}_p \)-Tucker's lemma seems to be new in the combinatorial literature.

Property (P), introduced in recent work and rooted in the classical theory of Parter vertices, concerns the existence of a nonsingular matrix \( A \in S(G) \) for which every vertex of \( G \) is a P-vertex. Previous investigations have fully characterized the property for trees, established it for cycles, extended it to unicyclic graphs, and shown that bipartite graphs with a perfect matching always satisfy property (P). However, whether the converse holds for connected bipartite graphs remains open in general.

In this paper, we make progress toward answering this question on multiple fronts. We first prove that every connected bipartite graph satisfying property (P) must be balanced, providing a fundamental necessary condition. We further establish complete characterizations for several significant families of bipartite graphs. Specially, we show that every connected bipartite graph of order at most \( 8 \) has property (P) if and only if it has a perfect matching, and more generally, that a connected bipartite graph of order \( 8+2k \) with at least \( k \) pendant edges satisfies property (P) exactly when it has a perfect matching. We further prove that within the class of triangular bipartite graphs, property (P) is equivalent to the existence of a perfect matching, providing a full characterization for this broad structural subclass.

The matroid complex, the simplicial complex of independent sets of a finite matroid, carries many significant invariants of the matroid by associating topological (\( T_0 \)-Alexandroff space) and algebraic structure (Stanley–Reisner ring) to the matroid. A central theme is to understand the homological invariants, in particular, the simplicial homology groups, Cohen–Macaulayness, and the graded Betti numbers. Foundational work of Reisner (1978) and Stanley (1976) explains how Cohen–Macaulay properties of Stanley–Reisner rings are obtained by the homology of links, while Hochster gives a direct formula relating graded Betti numbers to simplicial homology. On the topological side, Provan (1977) proved that matroid complexes are shellable, which implies that their reduced homology is concentrated in top dimension.

The \( q \)-matroids, which are a natural \( q \)-analogue of matroids, obtained by replacing subsets by vector subspaces, and in the representable case, they correspond to vector rank metric codes. In this talk, I will present the \( q \)-analogues of the above notions and results, based on a series of joint works [1, 2, 3] that initiate a homological framework for \( q \)-matroids. I will mention an application of these results to determining important discrete invariants of linear error-correcting codes, namely the higher weight spectra, in both Hamming and rank metrics.

[1] Shellability and Homology of \( q \)-Complexes and \( q \)-Matroids, Ghorpade, S., Pratihar, R., Randrianarisoa, T. H., Journal of Algebraic Combinatorics, 2022.

[2] The Euler Characteristic, \( q \)-Matroids and a Möbius Function, Johnsen, T., Pratihar, R., Randrianarisoa, T. H., Journal of Algebra and its Applications, 2025.

[3] Weight Spectra of Gabidulin Rank-Metric Codes and Betti Numbers, Johnsen, T., Pratihar, R., Verdure, H., São Paulo Journal of Mathematical Sciences, 2022.

For most problems related to perfect matchings, one may restrict attention to matching covered graphs — connected nonempty graphs in which each edge belongs to some perfect matching; Lucchesi and Murty's monograph [Perfect Matchings: A Theory of Matching Covered Graphs, 2024] discusses the extensive literature on them. A cornerstone of this theory is Lovász and Plummer's [Matching Theory, 1986] ear decomposition theorem, which yields interesting open problems in addition to being a fundamental problem-solving tool. We discuss two such open problems below.

By a bisubdivision of a graph \( J \), we mean a subdivision wherein each edge is replaced by a path of odd length; a connected graph (except \( K_2 \)) is matching covered if and only if each of its bisubdivisions is also matching covered. A subgraph \( H \) of a graph \( G \) is conformal if \( G - V(H) \) has a perfect matching. The aforementioned ear decomposition theorem of Lovász and Plummer implies that every matching covered graph (except \( K_2 \) and even cycles) has a conformal bisubdivision of either \( \theta \) or of \( K_4 \) (possibly both). This immediately leads to two problems: characterize \( \theta \)-free (matching covered) graphs, and likewise, \( K_4 \)-free ones. A characterization of planar \( K_4 \)-free matching covered graphs was obtained by Kothari and Murty (J. Graph Theory, 2016); the nonplanar case is open.

In the second Meru Combinatorics Conference (GEHU, Bhimtal, 2024), Rohinee Joshi presented a characterization of \( \theta \)-free graphs that relies heavily on a seminal result due to Edmonds, Lovász and Pulleyblank (Combinatorica, 1982) pertaining to the tight cut decomposition theory of matching covered graphs. As a sequel, we shall see some consequences of our characterization of \( \theta \)-free graphs, which includes an upper bound of \( 2n-2 \) on the size, forbidden minor type characterizations of Pfaffian (\( \theta \)-free) graphs and \( 3 \)-edge colorable cubic (\( \theta \)-free) graphs. We also prove that two seemingly unrelated classes of graphs — cubic \( \theta \)-free graphs and \( 2 \)-connected cubic graphs in which the length of each conformal cycle is divisible by four — turn out to be one and the same. The Petersen graph and \( K_4 \) play key roles in our results.

A nonempty connected graph \( G \) is matching covered if each edge participates in at least one perfect matching; an edge \( e \) is removable if \( G-e \) is also matching covered. There is extensive research on this graph class as documented in the recent monograph Perfect Matchings: A Theory of Matching Covered Graphs by Lucchesi and Murty (2024). In our on-going work, we investigate the relationship between two types of removable edges: b-invariant edges and solitary singletons.

A deep result of Lovász (JCT-B 1987) shows that every matching covered graph \( G \) admits a unique decomposition into special types called bricks and braces; \( b(G) \) denotes its number of bricks. A removable edge \( e \) is \( b \)-invariant if \( b(G-e)=b(G) \). Lovász conjectured that every brick, except \( K_4 \), triangular prism \( \overline{C_6} \), and the Petersen graph, has a \( b \)-invariant edge; this was proved by Carvalho, Lucchesi and Murty (JCT-B 2002).

An edge \( e \) is solitary if it participates in precisely one perfect matching. Recently, there has been renewed interest in them. An edge \( e \) depends on an edge \( f \) if every perfect matching containing \( e \) also contains \( f \); when restricted to solitary edges, this is an equivalence relation, and the corresponding equivalence classes are called solitary classes.

An edge \( e \) is a solitary singleton if it comprises a solitary class; it is easy to see that every solitary singleton is removable. We conjecture that every solitary singleton is in fact \( b \)-invariant. We have proved this for \( 3 \)-edge-connected \( r \)-graphs. This is joint work with Kothari.

The presentation is based on a paper that I co-authored with my supervisor. In the paper we have introduced the symmetric normaliser graph of a group \( G \). The vertex set of this graph consists of elements of the group. Vertices \( x \) and \( y \) are adjacent if \( x \) lies in the normaliser of \( \langle y \rangle \) and \( y \) lies in the normaliser of \( \langle x \rangle \). I shall discuss the hierarchical position this graph occupies in the hierarchy given by P J Cameron of graphs defined on groups. The existing hierarchy is further refined by this graph and the edges of this graph lie between the edges of the commuting graph and the nilpotent graph. For finite groups, I will discuss a necessary and sufficient condition for the symmetric normaliser graph to be equal to the commuting graph and similarly for equality with the nilpotent graph. Along with this, I will show the connections of this graph with the Engel graph and the non-generating graph of a group.

A graph is called a word-representable graph if there exists a word over its vertex set such that two vertices are adjacent in the graph if and only if they alternate in the word. The theory of word-representable graphs lies at the intersection of graph theory and combinatorics on words. This class of graphs contains several important classes such as circle graphs, 3-colorable graphs, and comparability graphs. In general, recognizing whether a graph is word-representable is an NP-complete problem. In this work, we characterize word-representable well-partitioned chordal (WPC) graphs using split decomposition. Further, we provide a polynomial-time recognition algorithm for word-representability of WPC graphs. The class of WPC graphs generalizes split graphs and is a subclass of chordal graphs. We also obtain the representation number of a word-representable WPC graph.

Ramsey-type problems for linear equations began with Schur’s theorem and were systematically generalized by Richard Rado. In the off-diagonal framework for two colors, one considers two different linear equations \( E_1, E_2 \) and determines the minimum integer \( N \) for which any red–blue coloring of \( \{1,2,\dots,N\} \) forces either a red solution of the equation \( E_1 \) or a blue solution of the equation \( E_2 \).

In this work, we study the off-diagonal Rado numbers for a combination of homogeneous and heterogeneous linear equations of the forms \( x+y+c=z \) and \( x+qy=z \). We determine the exact two-color off-diagonal Rado number \( R_2(c,q) \) associated with this system of equations.