Weisfeiler--Leman and Graph Spectra

We devise a hierarchy of spectral graph invariants, generalising the adjacency spectra and Laplacian spectra, which are commensurate in power with the hierarchy of combinatorial graph invariants generated by the Weisfeiler--Leman (WL) algorithm. More precisely, we provide a spectral characterisation of $k$-WL indistinguishability after $d$ iterations, for $k,d \in \mathbb{N}$. Most of the well-known spectral graph invariants such as adjacency or Laplacian spectra lie in the regime between 1-WL and 2-WL. We show that individualising one vertex plus running 1-WL is already more powerful than all such spectral invariants in terms of their ability to distinguish non-isomorphic graphs. Building on this result, we resolve an open problem of F\"urer (2010) about spectral invariants and strengthen a result due to Godsil (1981) about commute distances

The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

In this paper we study the complexity of the following problems: 1. Given a colored graph X=(V,E,c), compute a minimum cardinality set of vertices S (subset of V) such that no nontrivial automorphism of X fixes all vertices in S. A closely related problem is computing a minimum base S for a permutation group G <= S_n given by generators, i.e., a minimum cardinality subset S of [n] such that no nontrivial permutation in G fixes all elements of S. Our focus is mainly on the parameterized complexity of these problems. We show that when k=|S| is treated as parameter, then both problems are MINI[1]-hard. For the dual problems, where k=n-|S| is the parameter, we give FPT~algorithms. 2. A notion closely related to fixing is called individualization. Individualization combined with the Weisfeiler-Leman procedure is a fundamental technique in algorithms for Graph Isomorphism. Motivated by the power of individualization, in the present paper we explore the complexity of individualization: what is the minimum number of vertices we need to individualize in a given graph such that color refinement "succeeds" on it. Here "succeeds" could have different interpretations, and we consider the following: It could mean the individualized graph becomes: (a) discrete, (b) amenable, (c)compact, or (d) refinable. In particular, we study the parameterized versions of these problems where the parameter is the number of vertices individualized. We show a dichotomy: For graphs with color classes of size at most 3 these problems can be solved in polynomial time, while starting from color class size 4 they become W[P]-hard

Homomorphism Tensors and Linear Equations

Lov\'asz (1967) showed that two graphs $G$ and $H$ are isomorphic if and only if they are homomorphism indistinguishable over the class of all graphs, i.e. for every graph $F$, the number of homomorphisms from $F$ to $G$ equals the number of homomorphisms from $F$ to $H$. Recently, homomorphism indistinguishability over restricted classes of graphs such as bounded treewidth, bounded treedepth and planar graphs, has emerged as a surprisingly powerful framework for capturing diverse equivalence relations on graphs arising from logical equivalence and algebraic equation systems. In this paper, we provide a unified algebraic framework for such results by examining the linear-algebraic and representation-theoretic structure of tensors counting homomorphisms from labelled graphs. The existence of certain linear transformations between such homomorphism tensor subspaces can be interpreted both as homomorphism indistinguishability over a graph class and as feasibility of an equational system. Following this framework, we obtain characterisations of homomorphism indistinguishability over two natural graph classes, namely trees of bounded degree and graphs of bounded pathwidth, answering a question of Dell et al. (2018).Comment: 33 pages, accepted for ICALP 202

The Complexity of Homomorphism Indistinguishability

For every graph class {F}, let HomInd({F}) be the problem of deciding whether two given graphs are homomorphism-indistinguishable over {F}, i.e., for every graph F in {F}, the number hom(F, G) of homomorphisms from F to G equals the corresponding number hom(F, H) for H. For several natural graph classes (such as paths, trees, bounded treewidth graphs), homomorphism-indistinguishability over the class has an efficient structural characterization, resulting in polynomial time solvability [H. Dell et al., 2018]. In particular, it is known that two non-isomorphic graphs are homomorphism-indistinguishable over the class {T}_k of graphs of treewidth k if and only if they are not distinguished by k-dimensional Weisfeiler-Leman algorithm, a central heuristic for isomorphism testing: this characterization implies a polynomial time algorithm for HomInd({T}_k), for every fixed k in N. In this paper, we show that there is a polynomial-time-decidable class {F} of undirected graphs of bounded treewidth such that HomInd({F}) is undecidable. Our second hardness result concerns the class {K} of complete graphs. We show that HomInd({K}) is co-NP-hard, and in fact, we show completeness for the class C_=P (under P-time Turing reductions). On the algorithmic side, we show that HomInd({P}) can be solved in polynomial time for the class {P} of directed paths. We end with a brief study of two variants of the HomInd({F}) problem: (a) the problem of lexographic-comparison of homomorphism numbers of two graphs, and (b) the problem of computing certain distance-measures (defined via homomorphism numbers) between two graphs

CO to CO2 Using Magnesium Based Catalysts: An Overview

Stringent environmental re gulations have been adopted by the government in order to decrease the emission of vehicular exhaust such as S o x, N o x, C o and unb urned hydrocarbons. t herefore, the de velopment and exploration of catalysts started in the last century for the oxidation of carbon monoxide by different methods have attracted many researchers. t herefore, lar ge number of catalysts have been modified and tested for C o oxidation. t he de veloped catalysts have the ability of 100% conversion. Keeping in view of the literature accumulated in the last few decades for C o oxidation, Magnesium based catalysts ha ve been reported by many scientists for C o oxidation due to its unique characteristics such as high catalytic performance at low temperatures and good durability and stability toward C o oxidation. t his article represents a short re view in tabular form which facilitates a quick view on compounds that have been reported with magnesium previously