Arithmetic Properties of Overpartition $k$-Tuples with Odd Parts

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We prove several congruences for overpartition $k$-tuples with odd parts modulo powers of $2$, we prove a few infinite family of congruences for overpartition triples with odd parts and for overpartition $k$-tuples with odd parts (for $k=4$ and for an odd $k$). In addition, we discuss density results for overpartition $k$-tuples with odd parts modulo powers of primes. Our results are proved using elementary means as well as properties of modular forms. (This is joint work with Hirakjyoti Das, Abhishek Sarma, and James Sellers.)

An involution on reduced words and signed descent length enumeration

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Signed enumeration allows one to pass from the statistics on the Symmetric groups to statistics on the Alternating subgroup. Various signed enumeration results are known such as for excedance, excedance length, descent etc. We give an sign reversing involution that shows that $$\sum_{\pi \in S_n} \mathrm{sgn}(\pi)t^{\mathrm{exc}(\pi)}p^{\mathrm{exclen}(\pi)}q^{\mathrm{desclen}(\pi)}=(1−tpq)^{n−1},$$ where $\mathrm{exc}$ is the excedance number, $\mathrm{exclen}$ is the excedance length and $\mathrm{desclen}$ is the descent length. Setting $p=q=1$ gives us a result of Mantaci, while setting $t=q=1$ gives us a result of Sivasubramanian. We also show that the fixed points of the involution map to Motzkin paths of height utmost 1 under the Foata-Zeilberger map. We extend the signed distribution result for descent length to the type B and type D Coxeter groups. Finally, as applications, we obtain a complete matching of the Bruhat order for $S_n$ and $B_n$ and we calculate the mean and variance the descent length statistic for permutations sampled from the alternating group at random. This is joint work with S. Sivasubramanian.

Some recent results on fractional intersecting families

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For a fraction $\theta = a/b$ in $(0,1)$, a family $\mathcal{F}$ of subsets of $[n]$ is called a "fractional $\theta$-intersecting family" if, for every pair of distinct sets $A, B \in \mathcal{F}$, we have $\lvert A \cap B \rvert = \theta \lvert A \rvert$ or $\theta \lvert B \rvert$. The natural extremal question is: How large can a $\theta$-intersecting family over $[n]$ be? This notion was introduced in Balachandran--Mathew--Mishra (Electron. J. Combin. 26 (2019), #P2.40), wherein they showed that $\lvert \mathcal{F} \rvert \leq O(n \log n)$, and they gave constructions of $\theta$-intersecting families of size at least $\Omega(n)$. The conjecture (which is still open) is that $\lvert \mathcal{F} \rvert \leq O(n)$ for any $\theta$-intersecting family $\mathcal{F}$ over $[n]$. In this talk, I will discuss some recent progress on this conjecture, and some related questions concerning ranks of certain matrix ensembles, tournaments, symmetric designs, and sunflowers.

Characterization of $\theta$-free Matching Covered Graphs

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A nontrivial connected graph is matching covered if each edge belongs to some perfect matching. For most problems pertaining to perfect matchings, one may restrict attention to matching covered graphs; thus, there is an extensive theory on them. A cornerstone of this theory is an ear decomposition result due to Lovász and Plummer. Their theorem is a fundamental problem-solving tool, and also yields interesting open problems; we discuss two such problems below, and we solve one of them. A subgraph $H$ of a graph $G$ is conformal if $G-V(H)$ has a perfect matching. This notion is intrinsically related to the aforementioned ear decomposition theorem --- which implies that each matching covered graph (apart from $K_2$ and even cycles) contains a conformal bisubdivision of $\theta$, or a conformal bisubdivision of $K_4$, possibly both. (Here, $\theta$ refers to the graph with two vertices joined by three edges.) This immediately leads to two problems: characterize $\theta$-free (likewise, $K_4$-free) matching covered graphs. A characterization of planar $K_4$-free matching covered graphs was obtained by Kothari and Murty $(2016)$; the nonplanar case is still open. We provide a complete characterization of $\theta$-free matching covered graphs --- that is closely tied with the tight cut decomposition theory of matching covered graphs.

Square roots and character table sum

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The column sums of the character table of a finite group are always integers, which may also be negative. However, for finite Weyl groups (eg. symmetric group) these are non-negative. This is due to Frobenius and Schur, who interpreted column sums in terms of number of square roots of respective conjugacy classes. In this talk, we use their result to prove the validity of an intriguing property for Weyl groups (applicable to most finite groups) "The first column sum ( which records the dimensions of the irreducibles) dominate the sum of rest columns". While this property holds for any abelian group, generalized dihedral, and quaternionic groups, there are also a few exceptions. If time permits, we shall discuss the asymptotical behavior of the character table sums and derive their generating functions. This is an ongoing work with Arvind Ayyer and Hiranya Kishore Dey.

Koszul Duality for Schur Algebras and $q$-Schur Algebras

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The category of degree $d$ polynomial representations of $GL_{n}(\mathbb{C})$ is isomorphic to the category of modules of the Schur algebra $S=S(n,d)$, $d\geq 0$. The $q$-Schur algebra $S_{q}(n,d)$ is a quantum analogue of Schur algebra $S(n,d)$ whose module theory essentially captures polynomial representations of the quantum general linear group $GL_{q}(n)$. Geetha, Prasad and Srivastava (Pac. J. Math., 2020) defined the Koszul duality functor $D:S-\text{Mod} \xrightarrow{} S-\text{Mod}$ by $D(M)=S^{-}\otimes_{S} M$. This functor takes the Weyl module $W_{\lambda}$ to $W_{\lambda^{'}}$, for any partition $\lambda$ of $d$ with atmost $n$ parts. We will see an analogue of the Koszul duality functor for the $q$-Schur algebra $S_{q}(n,d)$. This is part of an ongoing work with A.~Prasad.

Zero-sum Problems for Restricted Sequences

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Zero-sum problems are basically combinatorial in nature. It deals with the condition that ensures that a given sequence over a finite group has a zero-sum subsequence with some prescribed properties. There are many invariants associated with zero-sum problems. Some such invariants are the Davenport constant, the Erdős-Ginzburg-Ziv constant, and the Harborth constant. The original motivation for introducing such group invariant was to study the problem of non-unique factorization domain over number fields. The precise value of such group invariants even for any finite abelian group are still unknown. In this talk, I will be discussing zero-sum problems over in some particular category of sequences. Moreover, we will study the behaviour of zero-sum invariants for random sequences.

A new perspective on Aztec Diamond Theorem

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We introduce a new algebraic technique to enumerate the tiling of Aztec Diamond. The main idea is to understand the tiling problem of a region that is a union of two regions sharing a common boundary where the tiling enumeration is already known. In this process, we get a clearer picture of connection of Aztec Diamond with Alternating Sign matrices. This a joint work with Manjil Saikia.

Sign-balance of excedances over Parity Alternating Permutations

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A permutation is called parity alternating if its entries assume even and odd integers alternately. In 2010, Tanimoto proved that the number of even non-parity alternating permutations with $k$ descents is same as the number of odd non-parity alternating permutations with $k$ descents. The objective of this paper is two-fold. The first is to find out whether an excedance counterpart of Tanimoto's result is true. The second is to find out the sign-balance for parity alternating permutations with respect to the statistic excedance. The results of this paper can be considered as an applications of linear algebraic techniques in the area of combinatorics, which are obtained by constructing suitable matrices and connecting their determinants with the signed-excedance enumeration of parity alternating permutations. As an application of the signed excedance enumeration, we prove that when $n \equiv 2 \pmod 4$, the excedance enumerating polynomials over the even and odd parity alternating permutations starting with an odd integer in $\mathfrak{S}_n$ are gamma positive. The sign-balance of parity alternating permutations with respect to several other statistics are also obtained.

Tiling proofs of Jacobi triple product and Rogers-Ramanujan identities

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We use the tiling method to give elementary combinatorial proofs of some celebrated q-series identities, such as Jacobi triple product identity, Rogers-Ramanujan identities, and some identities of Rogers. We give a tiling proof of the q-binomial theorem and a tiling interpretation of the q-binomial coefficients. A new generalized k-product q-series identity is also obtained by employing the tiling method, wherein the generating function of the set of all possible tilings of a rectangular board is computed in two different ways to obtain the desired q-series identity. Several new recursive q-series identities were also established. The `tiling-method' holds promise for giving an aesthetically pleasing approach to prove old and new q-series identities.

Weights Of Modules Over Heisenberg Lie Algebras

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Weight-sets $\mathrm{wt} V$ of finite-dim. simple/ integrable modules $V$, over finite-dim. complex simple/ affine Lie algebras $\mathfrak{g}(A)$ - among Kac-Moody(KM) and Borcherds-Kac-Moody(BKM) $\mathfrak{g}(A)$, for BKM-Cartan matrix $A$ - are well-known (by Weyl character formula classically): via tableau-theory in type $A$ by Schur polynomials (monomials/tableau-fillings); all finite types via various path models (Littlemann, Lakshmibai-Seshadri,...); affine types via partition-overlaid-patterns (GT-patterns); etc. For such non-integrable simple $V$ (infinite-dim., as highest weights non-dominant integral), Khare and Dhillon~computed~$\mathrm{wt} V$ via: subgroups of symmetries in Weyl group $W$, complementary freeness.

Over BKM $\mathfrak{g}(A)$, $\mathrm{wt} V$ of even integrable $V$ (despite Weyl-Kac-Borcherds character formula) seem unknown. Recently, we determined $\mathrm{wt} V$ explicitly - interestingly as Weyl subgroup orbits - for all highest weight (BKM) $\mathfrak{g}(A)$-modules $V$, via:
1) novel concept of {\it holes} (for simple $V$, size-1 holes are simple reflections in $W$ preserving $\wt V$);
2) enlarging the notion of dominant integral weights, thereby new integrable $V$ (characters unknown).

Here, we shall introduce the ingredients in 1), 2) above; and: compute $\mathrm{wt} V$ for modules $V$, over $\mathfrak{g}(A)$ generated by 3-dim. Heisenberg Lie algebras ($A_{ii}=0$ $\forall$ $i$, or negative part in $\mathfrak{g}(A)$ ``close to polynomials''), which turns-out to be crucial, and combinatorial via holes- relying on descriptions of the complements of $\mathbb{Z}_{\geq 0}$-cones in finite rank $\mathbb{Z}_{\geq 0}$-lattices. (Joint works with Apoorva Khare (in KM case) and Souvik Pal).

Solitary edges in cubic graphs

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Extensive research has been conducted on $2$-connected cubic~(aka $3$-regular) graphs. Petersen (1891) proved that every $2$-connected cubic graph has a perfect matching; and Sch\""onberger (1934) showed that each edge in a $2$-connected cubic graph is part of some perfect matching. In light of these well-known results, we set out to characterize those $2$-connected cubic graphs in which every edge is part of at least two perfect matchings.

An edge is referred to as a {\em solitary edge} if it belongs to a unique perfect matching. Thus, we can restate our objective as characterizing all $2$-connected cubic graphs that do not contain any solitary edges. In case of $2$-connected, there are graphs with $\frac{n}{2}$ solitary edges, where $n$ is the number of vertices. The problem turns out to be more interesting in the case of $3$-connected graphs. We have successfully established a constant upper bound (namely, six) on the number of solitary edges in $3$-connected cubic graphs.\par A connected graph $G$ with two or more vertices is said to be {\em matching covered} if each edge is part of at least one perfect matching. The concept of dependency relationship between edges, in a matching covered graph, was formally introduced and studied by Carvalho, Lucchesi and Murty~(1999). In a matching covered graph $G$, the dependency relationship between edges $e$ and $f$ may be defined as follows: edge $e$ depends on edge $f$, denoted as $e \rightarrow f$, if every perfect matching that includes edge $e$ also includes edge $f$, and edges $e$ and $f$ are mutually dependent, denoted as $e \leftrightarrow f$, if both $e \rightarrow f$ and $f \rightarrow e$. It is worth noting that mutual dependence is an equivalence relation implies it partitions the edge set $E(G)$ into distinct equivalence classes. The parts of this partition are referred to as the {\em equivalence classes} of $G$. And we refer to an equivalence class, say $R$, as a {\em removable class} if $G-R$ is matching covered. We say any two edges in a matching covered graph are {\em mutually exclusive} if there is no perfect matching that contains both the edges. The set of solitary edges in a $3$-connected cubic graph may be partitioned into $3$ (possibly empty) parts so that any two edges in the same part are mutually dependent, any two edges in the distinct parts are mutually exclusive, and each part has cardinality at most two; we refer to part of cardinality one as {\em solitary singleton} and part of cardinality two as {\em solitary doubleton}. We provide a complete characterization of all cubic 3-connected graphs that have at least three solitary edges.\par This is joint work with Kalyani Gohokar (CMI), Ajit Diwan (IIT Bombay) and Nishad Kothari (IIT Madras). This work is not yet published.

The bipartite Laplacian matrix of a nonsingular tree

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Similar to the bipartite adjacency matrix (J. A. Bondy and U. S. R. Murty. Graph theory, Springer-Verlag London, New York, 2008), we define the bipartite distance matrix of a bipartite graph with a unique perfect matching. The bipartite distance matrix $B(G)$ of a bipartite graph $G$ with a unique perfect matching on $2p$ vertices is a $p\times p$ matrix whose $(i, j)$th entry is the distance between vertices $l_i$ and $r_j$, where $L:=\{l_1,\ldots,l_p\}$, $R:=\{r_1,\ldots,r_p\}$ is a vertex bipartition of $G$. We observe that $\det {B}(G)$ is always a multiple of $2^{p-1}$. Based on this observation, we define the {\em bipartite distance index} of $G$ as $\texttt{bd}(G):=\det {B}(G)/ (-2)^{p-1}$.

We demonstrate that the bipartite distance index of a nonsingular tree $T$ satisfies an interesting inclusion-exclusion type of principle at any matching edge of the tree. Furthermore, we fully characterize the bipartite distance index of a nonsingular tree $T$ by the structure of $T$ through what we term $f$-alternating sums.

The study of the inverse of the bipartite distance matrix leads to an unexpected generalization of the usual Laplacian matrix for a tree, referred to as the bipartite Laplacian matrix. This generalized Laplacian matrix is usually not symmetric but it shares many properties with the usual Laplacian matrix. We study some of the fundamental properties of the bipartite Laplacian matrix and compare them with those of the usual Laplacian matrix. Lastly, we present a combinatorial description of all minors of the bipartite Laplacian matrix for a nonsingular tree.

A New Proof of an Additive Inversion Formula.

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Inversion formulas are extremely useful in combinatorics and number theory. We present an additive version of the well-known Mobius inversion formula. This involves the interesting function $h(r)$ which is a sum over all non-negative integer solutions of $1 b_1+...+ (r+1)b_{r+1}= 2r, b_1+...+b_{r+1}=r$, that is, over all integer partitions of $2r$ into exactly $r$ parts. This formula is equivalent to inverting a triangular Toeplitz matrix, as proved by Merca. Our proof of this inversion formula is combinatorial and new. The proof is a nice application of the principle of strong mathematical induction. We also have an application of the formula to obtain an anologue of a Guinand's result.

Roots of identity in finite groups of Lie type

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Given an integer $M\geq 2$, we deploy the generating function techniques to compute the number of $M$-th roots of identity in some of the well-known finite groups of Lie type, more precisely for finite general linear groups, symplectic groups, orthogonal groups of all types and unitary groups over finite fields of odd characteristics.

Planar Cycle-Extendable Graphs

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A nontrivial connected graph $G$ is said to be matching covered if each edge is part of some perfect matching. Most of the problems in matching theory can be reduced to matching covered graphs. Such graphs are also known as $1$-extendable graphs because each edge extends to a perfect matching. In a similar fashion, we say that a matching covered graph $G$ is cycle-extendable if (either) perfect matching of each even cycle $Q$ extends to a perfect matching (of $G$). Equivalently, a matching covered graph is cycle-extendable if each even cycle can be expressed as a symmetric difference of two perfect matchings. Another motivation to work on cycle-extendable graphs arises from the ear decomposition theory of matching covered graphs (that is similar to the well-known ear decomposition theory of 2-connected graphs). From this viewpoint, a matching covered graph $G$ is cycle-extendable if and only if each even cycle extends to an ear decomposition of $G$. As of now, there is no known NP characterization of cycle-extendable graphs (in general). Recently, we obtained NP (as well as P) characterizations of planar cycle-extendable graphs. Xiaofeng Guo and Fuji Zhang (2004) obtained such a characterization for bipartite planar cycle-extendable graphs (wherein they refer to cycle-extendable graphs as $1$-cycle resonant graphs). We provide a complete characterization of nonbipartite planar cycle-extendable graphs in terms of four infinite families. This is joint work with Kapil Pause (CMI), Rajat Adak (IIT Hyderabad), Ajit Diwan (IIT Bombay) and Nishad Kothari (IIT Madras).

Castelnuovo-Mumford regularity of the closed neighborhood ideal of a graph

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Let G be a finite simple graph, and NI(G) denote the closed neighborhood ideal of G in a polynomial ring R. We shall prove that if G is a forest, then the Castelnuovo-Mumford regularity of R/NI(G) is the same as the matching number of G, thus proving a conjecture of Sharifan and Moradi in the affirmative. We'll also see that the matching number of G provides a lower bound for the Castelnuovo-Mumford regularity of R/NI(G) whenever G is a chordal graph, complement of a tree, complete bipartite graph, cycle graph, or a wheel graph. If time permits, we'll also discuss the relationship between these two invariants for two graph operations: the join and the corona product of graphs.

Aspects of the Davenport constant for abelian groups of higher rank

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For $r\in\mathbb{N}$ we define the $r$-wise Davenport constant of a group $G$, denoted by $D_r(G)$, to be the least length of a sequence with elements from the group $G$ that ensures that we always have $r$ disjoint zero-sum subsequences. For $r=1$, this reduces to the classical definition of Davenport constant, mentioned by K. Rogers in 1963 in the context of it being an important invariant of the ideal class group in the ring of integers of an algebraic number field and subsequent applications to cryptology. In this talk, we shall try to discover its behaviour for a certain class of groups of rank $\geq 3$, for which its value was not known before. In this joint work with Dr. Eshita Mazumdar, we also propose new and arguably thinner bounds for the $r$-wise Davenport constant of any finite abelian group. The talk shall sketch many interesting open problems in the area as we go.

Clebsch-Gordan coefficients for Macdonald Polynomials

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The type $SL_2$ Macdonald polynomials $P_m$ and $E_m$ are $q,t$-deformation of characters of $SL_2$ representations. We give compact formulas for the $E$-expansion of the product $E_\ell P_m$ and $P$-expansion of the product $P_\ell P_m$ by using techniques from double affine Hecke algebra. This is based on joint work with Arun Ram.

Subset profiles of endofunctions on finite sets

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Let $[n]=\{1,\ldots,n\}$ be the set of first $n$ positive integers and $\mu=(\mu_1,\ldots,\mu_k)$ be a partition of $n$. An endofunction is a function whose domain is equal to its codomain. Let $T$ be an endofunction on $[n]$. A subset $W$ of $[n]$ has $T$-profile $\mu$ if $W \cup TW \cup \cdots T^{i-1}W = \mu_1+\cdots+\mu_i$ for all $1 \leq i \leq k$. Let $\sigma(\mu;T)$ denote the number of subsets with $T$-profile $\mu$. We study $\sigma(\mu;T)$ for endofunctions whose associated graph is a path, cycle or a tree and give precise formulae when $T$ is either a path or a cycle. Let $e_\lambda$, $h_\lambda$ and $p_\lambda$ denote the elementary, complete homogeneous and power-sum symmetric functions, respectively. We prove that if $T$ is a permutation of cycle type $\lambda$, then $\sigma(\mu;T)$ is the coefficient of $e_{\mu'}$ when $p_\lambda$ is expressed in the elementary basis, up to a sign, while if $T$ consists of paths whose lengths are given by $\lambda$, then $\sigma(\mu;T)$ is the coefficient of $e_{\mu'}$ when $h_\lambda$ is expressed in the elementary basis, up to a sign. We further show that if $T$ is an endofunction with cycles of type $\lambda$ and paths of lengths $\nu$, then $\sigma(\mu;T)$ is the coefficient of $e_{\mu'}$ when $p_\lambda. h_\nu$ is expressed in the elementary basis, up to a sign. We show that when $T$ is a tree, determination of $\sigma(\mu;T)$ is an NP-hard problem.

On the order sequence of a group

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The order sequence of a finite group $G$ is the list of orders of elements of the group, arranged in non-decreasing order. Order sequences of groups of order $n$ are ordered by elementwise domination, forming a partially ordered set. We prove a number of results about this poset, among them the following. $\bullet$ Abelian groups are uniquely determined by their order sequences, and the poset of order sequences of abelian groups of order $p^n$ is naturally isomorphic to the (well-studied) poset of partitions of $n$ with its natural partial order. $\bullet$ If there exists a non-nilpotent group of order $n$, then there exists such a group whose order sequence is dominated by the order sequence of any nilpotent group of order $n$. $\bullet$ There is a product operation on finite ordered sequences, defined by forming all products and sorting them into non-decreasing order. The product of order sequences of groups $G$ and $H$ is the order sequence of a group if and only if $|G|$ and $|H|$ are coprime.

Subsequence frequency in binary words

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A binary word is a finite sequence with terms in $\{0, 1\}$. We say that a binary word $w$ contains $k$ occurrences of a word $p$ if there are $k$ distinct subsequences of $w$ that match $p$. In this context, we usually call $p$ a pattern. The numbers we study are of the form $B_{n, p}(k)$, which is the number of binary words of length $n$ that have exactly $k$ occurrences of the pattern $p$. We first obtain an explicit formula for $B_{n, p}(k)$ for small values of $k$. We then turn to the numbers $M_{n, p}$, the maximum number of occurrences of $p$ that a word of length $n$ can have (the maximum value of $k$ for which $B_{n, p}(k) \neq 0$). We compute these numbers for certain special patterns $p$ and also address the following question: For any $k < M_{n, p}$, can we always find a word of length $n$ that has $k$ occurrences of $p$? Finally, we end with an interesting conjecture and problems for future research. This is based on joint work with Anurag Singh.

On Graphs Defined on Groups

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Associating graphs with groups dates back to Arthur Cayley. Starting from Cayley graphs, the research on graphs defined on algebraic objects like groups, rings, vector spaces etc. forms an integral part of algebraic graph theory. In this talk, I am going to discuss about another such graph defined on group, called \textit{comaximal subgroup graph $\Gamma(G)$} of a group $G$. It is defined as a graph with vertex set is the collection of all non-trivial proper subgroups of $G$ and two vertices $H$ and $K$ are adjacent if $HK=G$. Though the definition allows the possibility of $G$ to be infinite, in this discussion, I will focus mainly on finite groups. I will discuss various graph parameters like diameter, connectedness, girth, bipartiteness etc. We will see how the properties of underlying group structure influence the properties of graph and vice-versa. I will also highlight some problems on realizability and graph isomorphisms, and some partial solutions to those questions in terms of properties of $G$.

The bunkbed conjecture for the random cluster model

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The bunkbed conjecture is a well known unsolved problem in percolation theory that predicts that two natural notions of distance in graph theory should have the same qualitative behaviour. It says for instance that in a random edge subgraph of the n-hypercube, the probability that two points are in the same connected component is monotonically decreasing in their hamming distance. We explore this problem in the more general context of the random cluster model, which includes percolation as a special case and prove a number of results. In particular, we show that we can use linear algebraic arguments to prove the bunkbed conjecture for certain limiting cases of the random cluster measure. Our work naturally leads to some new and interesting correlation inequalities that extend the well known Rayleigh inequalities for spanning tree measures. This is joint work with Arvind Ayyer and Svante Linusson.

On the spectrum of generalized H-join operation constrained by indexing maps

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Fix $m \in \mathbb N$. A new generalization of the $H$-join operation of a family of graphs $\{G_1, G_2, \dots, G_k\}$ constrained by indexing maps $I_1,I_2,\dots,I_k$ is introduced as $H_m$-join of graphs, where the maps $I_i:V(G_i)$ to $[m]$. Various spectra, including adjacency, Laplacian, and signless Laplacian spectra, of any graph $G$, which is a $H_m$-join of graphs is obtained by introducing the concept of $E$-main eigenvalues. More precisely, we deduce that in the case of adjacency spectra, there is an associated matrix $E_i$ of the graph $G_i$ such that a $E_i$-non-main eigenvalue of multiplicity $m_i$ of $A(G_i)$ carry forward as an eigenvalue for $A(G)$ with the same multiplicity $m_i$, while an $E_i$-main eigenvalue of multiplicity $m_i$ carry forward as an eigenvalue of $G$ with multiplicity at least $m_i - m$. As a consequence, Universal spectra of any $H$-constrained join is obtained and cospectral mates are found.

Qualitative Independence Hypergraphs

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For positive integers $n$, $g$, and $t$ such that $t \geq 2$, $n \geq g^t$, a $t$-qualitative independence hypergraph, $t$-$QI(n, g)$, is a $t$-uniform hypergraph where the vertices are partitions $\mathcal{P}_i=(P_{i}^1, P_{i}^2, \ldots, P_{i}^g)$ of an $n$-set into $g$ classes with the property that every class of the partition has size at least $g^{t-1}$, and a set of $t$ vertices $\{\mathcal{P}_{i_1}, \mathcal{P}_{i_2}, \ldots, \mathcal{P}_{i_t}\}$ forms a hyperedge if the intersection of any $t$ classes from different partitions is nonempty, that is, $P_{i_1}^{j_1}\cap P_{i_2}^{j_2}\cap \cdots \cap P_{i_t}^{j_t} \neq \emptyset$ for $1\leq j_1, j_2, \ldots, j_t \leq g$. The family of $t$-qualitative independence hypergraphs gives a significant characterization for the existence of covering arrays on hypergraphs, which are combinatorial designs used in the interaction testing of softwares. For a $t$-uniform hypergraph $H=(V, E)$, a covering array on $H$, denoted by $CA(n, H, g)$, is an $n\times |V|$ matrix with entries from $\mathbb{Z}_g$ such that for every $t$ columns forming a hyperedge, every ordered $t$-tuple from $\mathbb{Z}_g^t$ appears in at least one row. A covering array with the least possible number of rows is called optimal. For a hypergraph $H$ and positive integers $g$ and $n$, a $CA(n, H, g)$ exists if and only if there is a hypergraph homomorphism $H \to t\text{-}QI(n, g)$. The question is: What is the minimum $n$ that guarantees the existence of such a homomorphism? In 2005, K. Meagher and B. Stevens (Covering Arrays on Graphs, Journal of Combinatorial Theory, Series B) showed that there is no homomorphism $2\text{-}QI(n, 2) \to 2\text{-}QI(n-1, 2)$. We study the strong chromatic number of $3\text{-}QI(n, 2)$ and establish the non-existence of a hypergraph homomorphism $3\text{-}QI(n, 2)\to 3\text{-}QI(n-1, 2)$. Consequently, we prove that an optimal covering array on $3\text{-}QI(n, 2)$ has precisely $n$ rows. This is a joint work with Dr. Yasmeen Akhtar, BITS Pilani, K K Birla Goa Campus.

Proper $q$-caterpillars are distinguished by their Chromatic Symmetric Functions

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Stanley's Tree Isomorphism Conjecture posits that the chromatic symmetric function can distinguish non-isomorphic trees. While already established for caterpillars and other subclasses of trees, we prove the conjecture's validity for a new class of trees that generalize proper caterpillars, thus confirming the conjecture for a broader class of trees.

Hypergraph symmetries and spectra of hypergraph matrices

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Symmetries of Hypergraphs and Some Invariant Subspaces of Matrices Associated with Hypergraphs.( https://arxiv.org/abs/2212.08314 )

On the connectivity and spectral properties of super graphs defined on finite groups

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For a finite group $G$, let $B$ be an equivalence (equality, conjugacy or order) relation on $G$ and let $A$ be a (power, enhanced power or commuting) graph with vertex set $G$. The $B$ super $A$ graph is a simple graph with vertex set $G$ and two vertices are adjacent if either they are in the same $B$-equivalence class or there are elements in their $B$-equivalence classes that are adjacent in the original $A$ graph. The graph obtained by deleting the dominant vertices (adjacent to all other vertices) from a $B$ super $A$ graph is called the reduced $B$ super $A$ graph. In this presentation, for some pairs of $B$ super $A$ graphs, we characterize the finite groups for which a pair of graphs are equal. We also characterize the dominant vertices for the order super commuting graph $\Delta^o(G)$ of $G$ and prove that for $n\geq 4$ the identity element is the only dominant vertex of $\Delta^o(S_n)$ and $\Delta^o(A_n)$. We characterize the values of $n$ for which the reduced order super commuting graph $\Delta^o(S_n)^*$ of $S_n$ and the reduced order super commuting graph $\Delta^o(A_n)^*$ of $A_n$ are connected. We also prove that if $\Delta^o(S_n)^*$ (or $\Delta^o(A_n)^*$) is connected then the diameter is at most $3$ and show that the diameter is $3$ for many value of $n.$ Later we will introduce the notion of generalized join and compressed of a graph with respect to certain equivalence relation on the vertex set. With the help of this notion, we will find the adjacency and Laplacian spectrum of conjugacy and order super commuting graphs of dihedral group $D_{2n}\; (n\geq 3)$, generalized quaternion group $Q_{4m} \;(m\geq 2)$ and the nonabelian group $\mathbb Z_p \rtimes \mathbb Z_q$ of order $pq$, where $p$ and $q$ are distinct primes with $q|p-1$.

Minimum order of a graph with given algebraic degree

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The algebraic degree $Deg(G)$ of a graph $G$ is the dimension of the splitting field of the adjacency polynomial of $G$ over the field of rationals $\mathbb{Q}$. The notion of algebraic degree was introduced by Monius et al. in 2018 as a generalization of the integral graphs i.e., graphs whose spectrum consists entirely of integers. It can be shown that for every positive integer $d$, there exists a circulant graph with algebraic degree $d$. Let $N_d$ be the least positive integer such that there exists a graph of order $N_d$ having algebraic degree $d$. In this paper we compute $N_d$ for small values of $d$ and establish some bounds on $N_d$ for a general $d$.

Hook restriction coefficients

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Finding a combinatorial interpretation of the multiplicities of each irreducible representation of the symmetric group $S_n$ in the restriction of an irreducible polynomial representation of the general linear group $GL_n(\mathbb{C})$ remains an interesting open problem in algebraic combinatorics. In this talk/poster we interpret the multiplicities of all irreducible representations of $S_n$ in the restriction of an irreducible polynomial representation of $GL_n(\mathbb{C})$ indexed by a hook partition.

On Automorphism Group of a Family of Symmetric Graphs

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In this paper, we study the automorphism group and geodesic transitivity of a family of vertex-transitive graphs $H(n,k)$, introduced by Fu-Tao Hu et.al. in 2010. In the process, we address some naturally arising, unanswered questions from that paper.

Generalized G-Rook Brauer algebras and their homology

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In this paper, we generalize the G-Rook Brauer algebras and their subalgebras by allowing more structured diagrams. We introduce equivariance by labelling edges of a diagram with elements of a group G. We then study the homology of G-Rook brauer diagram algebras.