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In a previous work, the authors showed that the maximum number of infinitely long synchronous directed walks that never meet is equal to the dimension d$$ d $$ of the no‐meet matroid, namely the largest order of a collection of vertex‐disjoint cycles. Given w>d$$ w>d $$, we want to compute the meeting time of w$$ w $$ walks: the first time step tw$$ {t}_w $$ such that, given any set of w$$ w $$ walks, at least two of them must meet no later than tw$$ {t}_w $$. We precisely prove that the meeting time is at most n−w+2$$ n-w+2 $$, where n$$ n $$ is the number of vertices. A connection is established with a cops and robber game on directed graphs with helicopter cops and an invisible slow robber. The meeting time of w$$ w $$ walks equals the capture time in this game, when at most w−1$$ w-1 $$ capture attempts are allowed. While this capture time can be computed in polynomial time, we show that it is NP‐hard to compute the minimum number of cops h$$ h $$ needed to catch the robber. More insights are also given on the number h$$ h $$ and its relation to pathwidth and other graph parameters. Finally we analyze these game measures on digraph tensor products. [ABSTRACT FROM AUTHOR]

Let G n be a graph on n vertices and let w k (G n) denote the number of walks of length k in G n divided by n. Erdős and Simonovits conjectured that w k (G n) t ≥ w t (G n) k when k ≥ t and both t and k are odd. In 2018, Sağlam proved this conjecture. We give a new shorter proof of this result. [ABSTRACT FROM AUTHOR]

The paper is concerned with Paul Auster's novel Oracle Night (2003) from the point of view of the importance of the city and its analogy to human thoughts, feelings, and life itself. Acknowledging the relevant theoretical framework, such as Elizabeth Grosz's concept of 'interface,' Yi-Fu Tuan's notions of space and place, as well as various critical views on Auster's works, the paper aims to contrast "closed spaces" with "open-air city scenes," in order to emphasize that closed spaces mirror the main character's confusion, whereas open-air urban scenes reflect his increased clarity of vision. The paper concludes that the feelings of being alive and happy seem possible, certain, and hopefully permanent only in contact with the streets, outside all rooms, and free from locks. In addition, by analyzing the nature of writing, the paper finds that the main character is the embodiment of the interface (Grosz 248) between the private/body and the shared/city. [ABSTRACT FROM AUTHOR]

To the Woman of England: Charlotte Smith’s Rural Walks – Charlotte Smith proposes her children’s literature as a genre shaped by reflections on gender. The work Rural Walks depicts the mother figure as an educational mentor who nurtures and directs the growth of her young girls and as a heroine of daily life. Smith wants to give her female readers a revolutionary version of motherhood, for women are the primary recipients of Mrs. Woodfield’s lessons. While focusing on the mental and moral development of young women, she also attends to her own social and moral responsibilities. Her educational program is comprehensive of unusual fields of knowledge and addresses social issues. In her children’s books Smith instructs her female readers how to apply the social lessons of her literature to their experiences in the real world and how to become politically active citizens. In Rural Walks the dialogues illustrate the overlapping theories on the formation of the self and the education of the woman.

Let G be a graph with m edges and let ρ be the largest eigenvalue of its adjacency matrix. It is shown that

A properly colored walk in an edge-colored graph is a walk such that consecutive edges are of distinct colors. In this paper, based on a transformation from directed graphs to edge-colored graphs, we classified edge-colored graphs into three families: degenerate edge-colored graphs, semi-degenerate edge-colored graphs and non-degenerate graphs. By a polynomial-time computable parameter related to properly colored walks, we gave a characterization of these three families. Applying this characterization, we slightly strengthened Yeo's Theorem (Every edge-colored graph G containing no PC cycle contains a vertex z ∈ V (G) such that each component of G − z is joint to z with at most one color , Yeo, 1997). [ABSTRACT FROM AUTHOR]

We initiate the study of a fundamental combinatorial problem: Given a capacitated graph G = (V , E) , find a shortest walk ("route") from a source s ∈ V to a destination t ∈ V that includes all vertices specified by a set W P ⊆ V : the waypoints. This Waypoint Routing Problem finds immediate applications in the context of modern networked systems. Our main contribution is an exact polynomial-time algorithm for graphs of bounded treewidth. We also show that if the number of waypoints is logarithmically bounded, exact polynomial-time algorithms exist even for general graphs. Our two algorithms provide an almost complete characterization of what can be solved exactly in polynomial time: we show that more general problems (e.g., on grid graphs of maximum degree 3, with slightly more waypoints) are computationally intractable. [ABSTRACT FROM AUTHOR]

We provide an introduction to graph theory and linear algebra. The present article consists of two parts. In the first part, we review the transfer-matrix method. It is known that many enumeration problems can be reduced to counting walks in a graph. After recalling the basics of linear algebra, we count walks in a graph by using eigenvalues. In the second part, we introduce PageRank by using a random walk model. PageRank is a method to estimate the importance of web pages and is one of the most successful algorithms. This article is based on the author's lectures at Tohoku University in 2018 and 2020. [ABSTRACT FROM AUTHOR]

Since some experiments analyzed impact of navigation tools on pedestrian navigation (Ishikawa, Fujiwara, Imai & Okabe 2008; Wang & Worboys 2016) few have studied how interactive devices between mobile participants possibly transform their cognitive maps. After many years of interactive artistic experiences between participants in distant cities, the goal ofthe present research is a better understanding of the links between mental, instrumental and shared maps. The question is whether connected and dynamic applications renew our shared mental representations of urban spaces. We will approach the notion of mental maps (introduced by Tolman) in its individual and collective dimension with regard to the new usescreated by connecting devices. The hypothesis is that new access to cartographic tools is likely to produce new kinds of individual mental representations. Method: in these preliminary results, the objective is to compare the mental maps evidenced by maps drawn after the exploration of a single urban district between 2 groups of participants: 1) group of individuals equipped simply with a passive GPS tracking tool, 2) group of individuals equipped with an urban navigation application (Google map). Measures will include: 1) comparison of landmarks hierarchies, 2) Comparison of routes tracesand icons, 3) Direct comparison by distances between geographical landmarks and drawing landmarks by superposition of the two kinds of maps, 4) Relative comparison between internal distances in geographical landmarks and internal distances between drawings landmarks. We will compare the two groups relative to these measures. In these preliminary investigations, and contrary to some assumptions (Ishikawa, Wang) we cannot find obvious confirmation that pedestrian’susers of Google map have worse results from cognitive mapping that others without this device.

The edge reconstruction conjecture of Harary (1964) states that a finite graph G can be reconstructed up to isomorphism from the multiset of its edge-deleted subgraphs G - e (with e running over the edges of G). We put this conjecture in the framework of measure-theoretic rigidity, revealing the importance of the lengths of labeled closed walks for the problem. [ABSTRACT FROM AUTHOR]

Since some experiments analyzed impact of navigation tools on pedestrian navigation (Ishikawa, Fujiwara, Imai & Okabe 2008; Wang & Worboys 2016) few have studied how interactive devices between mobile participants possibly transform their cognitive maps. After many years of interactive artistic experiences between participants in distant cities, the goal of the present research is a better understanding of the links between mental, instrumental and shared maps. The question is whether connected and dynamic applications renew our shared mental representations of urban spaces. We will approach the notion of mental maps (introduced by Tolman) in its individual and collective dimension with regard to the new uses created by connecting devices. The hypothesis is that new access to cartographic tools is likely to produce new kinds of individual mental representations. Method: in these preliminary results, the objective is to compare the mental maps evidenced by maps drawn after the exploration of a single urban district between 2 groups of participants: 1) group of individuals equipped simply with a passive GPS tracking tool, 2) group of individuals equipped with an urban navigation application (Google map). Measures will include: 1) comparison of landmarks hierarchies, 2) Comparison of routes traces and icons, 3) Direct comparison by distances between geographical landmarks and drawing landmarks by superposition of the two kinds of maps, 4) Relative comparison between internal distances in geographical landmarks and internal distances between drawings landmarks. We will compare the two groups relative to these measures. In these preliminary investigations, and contrary to some assumptions (Ishikawa, Wang) we cannot find obvious confirmation that pedestrian's users of Google map have worse results from cognitive mapping that others without this device. [ABSTRACT FROM AUTHOR]

This essay creates an alternative document to the performative artwork HOW - Heathrow Orchard Walks (2014-2016). HOW was devised and conducted by the author Kate Corder and co-performed knowingly or unknowingly by participants. Through a series of research walks HOW investigates, explores and experiences contested land surrounding Heathrow Airport. Notions pertaining to airport expansion and "third runway" suggestions threaten the area perpetuating uncertainty in local communities and wider populations. Undesirable losses of habitat, continual environmental damage and intensifying pollution would accompany the airport’s growth. The Government's Davies Commission Report published in July 2015 aggravated this situation further by advocating Heathrow's viability for airport expansion. Understanding histories of cultivation at Heathrow was essential to formulate HOW activity in establishing paths and destinations. Previously market gardens and orchards in the region grew quantities of fruit and vegetables supplying London; currently, most orchards have disappeared, arable crops are grown and gravel excavations erode the soil. During HOW participants and routes varied generating a layering of happenings built over time as the series evolved. The tour guide was the consistent still point within the shared experience of the activity. Photographic images taken during HOW transmit the transient nature of walking in a contested landscape. [ABSTRACT FROM AUTHOR]

The diagonal of a multivariate power series F is the univariate power series Diag F generated by the diagonal terms of F . Diagonals form an important class of power series; they occur frequently in number theory, theoretical physics and enumerative combinatorics. We study algorithmic questions related to diagonals in the case where F is the Taylor expansion of a bivariate rational function. It is classical that in this case Diag F is an algebraic function. We propose an algorithm that computes an annihilating polynomial for Diag F . We give a precise bound on the size of this polynomial and show that, generically, this polynomial is the minimal polynomial and that its size reaches the bound. The algorithm runs in time quasi-linear in this bound, which grows exponentially with the degree of the input rational function. We then address the related problem of enumerating directed lattice walks. The insight given by our study leads to a new method for expanding the generating power series of bridges, excursions and meanders. We show that their first N terms can be computed in quasi-linear complexity in N , without first computing a very large polynomial equation. [ABSTRACT FROM AUTHOR]

Degree assortativity is the tendency for nodes of high degree (resp. low degree) in a graph to be connected to high degree nodes (resp. to low degree ones). It is usually quantified by the Pearson correlation coefficient of the degree–degree correlation. Here we extend this concept to account for the effect of second neighbours to a given node in a graph. That is, we consider the two-walks degree of a node as the sum of all the degrees of its adjacent nodes. The two-walks degree assortativity of a graph is then the Pearson correlation coefficient of the two-walks degree–degree correlation. We found here analytical expression for this two-walks degree assortativity index as a function of contributing subgraphs. We then study all the 261,000 connected graphs with 9 nodes and observe the existence of assortative–assortative and disassortative–disassortative graphs according to degree and two-walks degree, respectively. More surprisingly, we observe a class of graphs which are degree disassortative and two-walks degree assortative. We explain the existence of some of these graphs due to the presence of certain topological features, such as a node of low-degree connected to high-degree ones. More importantly, we study a series of 49 real-world networks, where we observe the existence of the disassortative–assortative class in several of them. In particular, all biological networks studied here were in this class. We also conclude that no graphs/networks are possible with assortative–disassortative structure. [ABSTRACT FROM AUTHOR]

Partially commutative monoids provide a powerful tool to study graphs, viewing walks as words whose letters, the edges of the graph, obey a specific commutation rule. A particular class of traces emerges from this framework, the hikes, whose alphabet is the set of simple cycles on the graph. We show that hikes characterize undirected graphs uniquely, up to isomorphism, and satisfy remarkable algebraic properties such as the existence and uniqueness of a prime factorization. Because of this, the set of hikes partially ordered by divisibility hosts a plethora of relations in direct correspondence with those found in number theory. Some applications of these results are presented, including a permanantal extension to MacMahon's master theorem and a derivation of the Ihara zeta function. [ABSTRACT FROM AUTHOR]

A nonlinear random walk related to the porous medium equation (nonlinear Fokker-Planck equation) is investigated. This random walk is such that when the number of steps is sufficiently large, the probability of finding the walker in a certain position after taking a determined number of steps approximates to a q-Gaussian distribution (Gq,β(x) ∝ [1 - (1 - q)βx²]1/(1-q)), which is a solution of the porous medium equation. This can be seen as a verification of a generalized central limit theorem where the attractor is a q-Gaussian distribution, reducing to the Gaussian one when the linearity is recovered (q → 1). In addition, motivated by this random walk, a nonlinear Markov chain is suggested. [ABSTRACT FROM AUTHOR]

A segment of a tree T is a path whose end vertices have degree 1 or at least 3, while all internal vertices have degree 2. The lengths of all the segments of T form its segment sequence, in analogy to the degree sequence. We address a number of extremal problems for the class of all trees with a given segment sequence. In particular, we determine the extremal trees for the number of subtrees, the number of matchings and independent sets, the graph energy, and spectral moments. [ABSTRACT FROM AUTHOR]

Let Z n be the ring of residue classes modulo n , and let Z n ⁎ be the group of its units. 90 years ago, Brauer obtained a formula for the number of representations of c ∈ Z n as the sum of k units. Recently, Yang and Tang (2015) [6] gave a formula for the number of solutions of the equation x 1 2 + x 2 2 = c with x 1 , x 2 ∈ Z n ⁎ . In this paper, we generalize this result. We find an explicit formula for the number of solutions of the equation x 1 2 + ⋯ + x k 2 = c with x 1 , … , x k ∈ Z n ⁎ . [ABSTRACT FROM AUTHOR]

David Finkelstein was very fond of the new information-theoretic paradigm of physics advocated by John Archibald Wheeler and Richard Feynman. Only recently, however, the paradigm has concretely shown its full power, with the derivation of quantum theory (Chiribella et al., Phys. Rev. A 84:012311, 2011; D'Ariano et al., 2017) and of free quantum field theory (D'Ariano and Perinotti, Phys. Rev. A 90:062106, 2014; Bisio et al., Phys. Rev. A 88:032301, 2013; Bisio et al., Ann. Phys. 354:244, 2015; Bisio et al., Ann. Phys. 368:177, 2016) from informational principles. The paradigm has opened for the first time the possibility of avoiding physical primitives in the axioms of the physical theory, allowing a re-foundation of the whole physics over logically solid grounds. In addition to such methodological value, the new information-theoretic derivation of quantum field theory is particularly interesting for establishing a theoretical framework for quantum gravity, with the idea of obtaining gravity itself as emergent from the quantum information processing, as also suggested by the role played by information in the holographic principle (Susskind, J. Math. Phys. 36:6377, 1995; Bousso, Rev. Mod. Phys. 74:825, 2002). In this paper I review how free quantum field theory is derived without using mechanical primitives, including space-time, special relativity, Hamiltonians, and quantization rules. The theory is simply provided by the simplest quantum algorithm encompassing a countable set of quantum systems whose network of interactions satisfies the three following simple principles: homogeneity, locality, and isotropy. The inherent discrete nature of the informational derivation leads to an extension of quantum field theory in terms of a quantum cellular automata and quantum walks. A simple heuristic argument sets the scale to the Planck one, and the currently observed regime where discreteness is not visible is the so-called 'relativistic regime' of small wavevectors, which holds for all energies ever tested (and even much larger), where the usual free quantum field theory is perfectly recovered. In the present quantum discrete theory Einstein relativity principle can be restated without using space-time in terms of invariance of the eigenvalue equation of the automaton/walk under change of representations. Distortions of the Poincaré group emerge at the Planck scale, whereas special relativity is perfectly recovered in the relativistic regime. Discreteness, on the other hand, has some plus compared to the continuum theory: 1) it contains it as a special regime; 2) it leads to some additional features with GR flavor: the existence of an upper bound for the particle mass (with physical interpretation as the Planck mass), and a global De Sitter invariance; 3) it provides its own physical standards for space, time, and mass within a purely mathematical adimensional context. The paper ends with the future perspectives of this project, and with an Appendix containing biographic notes about my friendship with David Finkelstein, to whom this paper is dedicated. [ABSTRACT FROM AUTHOR]

In the article is presented the complex rehabilitation program for pregnant with alimentary-constitutional obesity in the third trimester. The program is directed on the decline of bodyweight, with the purpose of increase of scray-protective reactions of organism of pregnant and to the decline of obstetric traumatism in births. In an experiment took part 90 pregnant of the third trimester with obesity of I-III degree, middle genesial age. The developed and inculcated program foresaw the use of physical, psychological and medical rehabilitation.

During the last decades there is a growing interest in children's experiences of their everyday environments, both in research and pedagogical practices. Research methods that enable children to participate actively in research processes have been developed. One such method is based on walks, in which children guide a researcher in their everyday environments. The present article aims to analyze experiences from such walks. The walks were part of a project where children’s relations to outdoor places, in nature environments and in schoolyards, were explored. 41 children, aged six to eight years, participated in 39 videotaped walks. The children photographed with digital cameras during the walks. The analysis shows that walks enable children to express themselves verbally as well as non-verbally, by telling and demonstrating what can be done in the different places. Embodied experiences are manifested when children interplay with the environment and memories are evoked when they catch sight of objects and places. Furthermore, children are empowered by deciding the route and what they chose to show. The method is useful both for researchers and practitioners in order to know how children experience their everyday environment.

Abstract: Let be the eigenvalues of a graph G. For any , the k-th spectral moment of G is defined by . We use the fact that is also the number of closed walks of length k in G to show that among trees T whose degree sequence is D or majorized by D, is maximized by the greedy tree with degree sequence D (constructed by assigning the highest degree in D to the root, the second-, third-, …highest degrees to the neighbors of the root, and so on) for any . Several corollaries follow, in particular a conjecture of Ilić and Stevanović on trees with given maximum degree, which in turn implies a conjecture of Gutman, Furtula, Marković and Glišić on the Estrada index of such trees, which is defined as . [Copyright &y& Elsevier]

In this article the phenomenon of literary tourism is approached through the Foucauldian concept heterotopias. A heterotopia is a bridge between worlds, a physical place or a representation of a place, where transgression is possible. This can be applied to literary tourism, where the tourist goes between his or her own experiences of reading, into the more public experience which is formulated through guide books and other enunciations of the tourist discourse. The main example is a series of literary walks in Cornwall, connected to the novelist Daphne du Maurier. The literary experience is related to the physical experience and the corporeality of the reading act.

Graph kernels are powerful tools to bridge the gap between machine learning and data encoded as graphs. Most graph kernels are based on the decomposition of graphs into a set of patterns. The similarity between two graphs is then deduced to the similarity between corresponding patterns. Kernels based on linear patterns constitute a good trade-off between accuracy and computational complexity. In this work, we propose a thorough investigation and comparison of graph kernels based on different linear patterns, namely walks and paths. First, all these kernels are explored in detail, including their mathematical foundations, structures of patterns and computational complexity. After that, experiments are performed on various benchmark datasets exhibiting different types of graphs, including labeled and unlabeled graphs, graphs with different numbers of vertices, graphs with different average vertex degrees, linear and non-linear graphs. Finally, for regression and classification tasks, accuracy and computational complexity of these kernels are compared and analyzed, in the light of baseline kernels based on non-linear patterns. Suggestions are proposed to choose kernels according to the types of graph datasets. This work leads to a clear comparison of strengths and weaknesses of these kernels. An open-source Python library containing an implementation of all discussed kernels is publicly available on GitHub to the community, thus allowing to promote and facilitate the use of graph kernels in machine learning problems. • Graph kernels are powerful to bridge the gap between machine learning and graphs. • Graph kernels based on linear patterns can achieve good performances. • Graph kernels should be chosen carefully according to the types of graph datasets. [ABSTRACT FROM AUTHOR]

This study presents military training areas as important military landscapes, and sites of conflict in their own right. It explores the development of military-environmentalism at two active British training areas, Salisbury Plain Training Area and Lulworth Range, Dorset. It contrasts the rise of the military-environmental discourse with the removal of human inhabitants of the areas during the Second World War. The public protests and memorial attempts that followed, it argues, along with (but not allied with) emerging environmentalism, shaped military understanding of the value of its landholdings. Two walks at Salisbury Plain Training Area and Imber highlight the critical role of access in militarized landscapes. This study goes beyond the barbed wire to reveal military training areas as complex places, at which a site-based perspective teases out the nuances of broad narratives-war, rural development, environmentalism, and anti-military protest-as they have played out on two different landscapes from 1943 to the present. [ABSTRACT FROM AUTHOR]

Abstract: We study a generalization of strongly regular graphs. We call a graph strongly walk-regular if there is an such that the number of walks of length ℓ from a vertex to another vertex depends only on whether the two vertices are the same, adjacent, or not adjacent. We will show that a strongly walk-regular graph must be an empty graph, a complete graph, a strongly regular graph, a disjoint union of complete bipartite graphs of the same size and isolated vertices, or a regular graph with four eigenvalues. Graphs from the first three families in this list are indeed strongly ℓ-walk-regular for all ℓ, whereas the graphs from the fourth family are ℓ-walk-regular for every odd ℓ. The case of regular graphs with four eigenvalues is the most interesting (and complicated) one. Such graphs cannot be strongly ℓ-walk-regular for even ℓ. We will characterize the case that regular four-eigenvalue graphs are strongly ℓ-walk-regular for every odd ℓ, in terms of the eigenvalues. There are several examples of infinite families of such graphs. We will show that every other regular four-eigenvalue graph can be strongly ℓ-walk-regular for at most one ℓ. There are several examples of infinite families of such graphs that are strongly 3-walk-regular. It however remains open whether there are any graphs that are strongly ℓ-walk-regular for only one particular ℓ different from 3. [Copyright &y& Elsevier]

Let X be a graph on ν vertices with adjacency matrix A, and let S be a subset of its vertices with characteristic vector z. We say that the pair ( X, S) is controllable if the vectors A z for r = 1, . . . , ν − 1 span $${\mathbb{R}^{\nu}}$$ . Our concern is chiefly with the cases where S = V( X), or S is a single vertex. In this paper we develop the basic theory of controllable pairs. We will see that if ( X, S) is controllable then the only automorphism of X that fixes S as a set is the identity. If ( X, S) is controllable for some subset S then the eigenvalues of A are all simple. [ABSTRACT FROM AUTHOR]

Symbolic dynamics is partly the study of walks in a directed graph. By a walk, here we mean a morphism to the graph from the Cayley graph of the monoid of non-negative integers. Sets of these walks are also important in other areas, such as stochastic processes, automata, combinatorial group theory, C"-algebras, etc. We put a Quillen model structure on the category of directed graphs, for which the weak equivalences are those graph morphisms which induce bijections on the set of walks. We determine the resulting homotopy category. We also introduce a "finite-level" homotopy category which respects the natural topology on the set of walks. To each graph we associate a basal graph, well defined up to isomorphism. We show that the basal graph is a homotopy invariant for our model structure, and that it is a finer invariant than the zeta series of a finite graph. We also show that, for finite walkable graphs, if B is basal and separated then the walk spaces for X and B are topologically conjugate if and only if X and B are homotopically equivalent for our model structure [ABSTRACT FROM AUTHOR]

We present a computer-aided, yet fully rigorous, proof of Ira Gessel's tantalizingly simply stated conjecture that the number of ways of walking 2n steps in the region x +y ≥ 0,y ≥ 0 of the square lattice with unit steps in the east, west, north, and south directions, that start and end at the origin, equals (Multiple line equation(s) cannot be represented in ASCII text). [ABSTRACT FROM AUTHOR]

This article shows how the panorama, as a modern form of visualization and representation, became a model for the Santa Lucia Hill and the Cousino Park, two emblematic projects of Santiago's transformation, carried out by Benjamin Vicuna Mackenna between 1872 and 1875. On the one hand, supported by the height and location of the hill and its pattern of paths and terraces, the Santa Lucia promenade became an effective mechanism for visual control of the city as a whole, as well as a reference to the new urban order that was to be imposed. On the other hand, by proposing a landscape design that served as an urban window into Santiago's geographical surroundings, the Cousino Park project interpreted the qualities of the panorama from a strongly local sensitivity that highlighted the values of life in contact with nature. The analysis of these two cases will show how the panorama remarkably influenced Santiago's first transformation in the republican era, while representing fundamental values of the emerging national identity. [ABSTRACT FROM AUTHOR]

In this paper, we derive the generating function of RNA structures with pseudoknots. We enumerate all k-noncrossing RNA pseudoknot structures categorized by their maximal sets of mutually intersecting arcs. In addition, we enumerate pseudoknot structures over circular RNA. For 3-noncrossing RNA structures and RNA secondary structures we present a novel 4-term recursion formula and a 2-term recursion, respectively. Furthermore, we enumerate for arbitrary k all k-noncrossing, restricted RNA structures i.e. k-noncrossing RNA structures without 2-arcs i.e. arcs of the form (i, i + 2), for 1 ≤ i ≤ n-2. [ABSTRACT FROM AUTHOR]

Abstract: We consider symmetric powers of a graph. In particular, we show that the spectra of the symmetric square of strongly regular graphs with the same parameters are equal. We also provide some bounds on the spectra of the symmetric squares of more general graphs. The connection with generic exchange Hamiltonians in quantum mechanics is discussed in Appendix A. [Copyright &y& Elsevier]