On the spectral moments of trees with a given bipartition

For two given positive integers $p$ and $q$ with $p\leqslant q$, we denote \mathscr{T}_n^{p, q}={T: T is a tree of order $n$ with a $(p, q)$-bipartition}. For a graph $G$ with $n$ vertices, let $A(G)$ be its adjacency matrix with eigenvalues $\lambda_1(G), \lambda_2(G), ..., \lambda_n(G)$ in non-increasing order. The number $S_k(G):=\sum_{i=1}^{n}\lambda_i^k(G)\,(k=0, 1, ..., n-1)$ is called the $k$th spectral moment of $G$. Let $S(G)=(S_0(G), S_1(G),..., S_{n-1}(G))$ be the sequence of spectral moments of $G$. For two graphs $G_1$ and $G_2$, one has $G_1\prec_s G_2$ if for some $k\in {1,2,...,n-1}$, $S_i(G_1)=S_i(G_2) (i=0,1,...,k-1)$ and $S_k(G_1)<S_k(G_2)$ holds. In this paper, the last four trees, in the $S$-order, among $\mathscr{T}_n^{p, q} (4\leqslant p\leqslant q)$ are characterized.Comment: 11 pages, 7 figure

Institutional Cognition

We generalize a recent mathematical analysis of Bernard Baars' model of human consciousness to explore analogous, but far more complicated, phenomena of institutional cognition. Individual consciousness is limited to a single, tunable, giant component of interacting cogntivie modules, instantiating a Global Workspace. Human institutions, by contrast, seem able to multitask, supporting several such giant components simultaneously, although their behavior remains constrained to a topology generated by cultural context and by the path-dependence inherent to organizational history. Surprisingly, such multitasking, while clearly limiting the phenomenon of inattentional blindness, does not eliminate it. This suggests that organizations (or machines) explicitly designed along these principles, while highly efficient at certain sets of tasks, would still be subject to analogs of the subtle failure patterns explored in Wallace (2005b, 2006). We compare and contrast our results with recent work on collective efficacy and collective consciousness

Darwin's Rainbow: Evolutionary radiation and the spectrum of consciousness

Evolution is littered with paraphyletic convergences: many roads lead to functional Romes. We propose here another example - an equivalence class structure factoring the broad realm of possible realizations of the Baars Global Workspace consciousness model. The construction suggests many different physiological systems can support rapidly shifting, sometimes highly tunable, temporary assemblages of interacting unconscious cognitive modules. The discovery implies various animal taxa exhibiting behaviors we broadly recognize as conscious are, in fact, simply expressing different forms of the same underlying phenomenon. Mathematically, we find much slower, and even multiple simultaneous, versions of the basic structure can operate over very long timescales, a kind of paraconsciousness often ascribed to group phenomena. The variety of possibilities, a veritable rainbow, suggests minds today may be only a small surviving fraction of ancient evolutionary radiations - bush phylogenies of consciousness and paraconsciousness. Under this scenario, the resulting diversity was subsequently pruned by selection and chance extinction. Though few traces of the radiation may be found in the direct fossil record, exaptations and vestiges are scattered across the living mind. Humans, for instance, display an uncommonly profound synergism between individual consciousness and their embedding cultural heritages, enabling efficient Lamarkian adaptation

Machine Hyperconsciousness

Individual animal consciousness appears limited to a single giant component of interacting cognitive modules, instantiating a shifting, highly tunable, Global Workspace. Human institutions, by contrast, can support several, often many, such giant components simultaneously, although they generally function far more slowly than the minds of the individuals who compose them. Machines having multiple global workspaces -- hyperconscious machines -- should, however, be able to operate at the few hundred milliseconds characteistic of individual consciousness. Such multitasking -- machine or institutional -- while clearly limiting the phenomenon of inattentional blindness, does not eliminate it, and introduces characteristic failure modes involving the distortion of information sent between global workspaces. This suggests that machines explicitly designed along these principles, while highly efficient at certain sets of tasks, remain subject to canonical and idiosyncratic failure patterns analogous to, but more complicated than, those explored in Wallace (2006a). By contrast, institutions, facing similar challenges, are usually deeply embedded in a highly stabilizing cultural matrix of law, custom, and tradition which has evolved over many centuries. Parallel development of analogous engineering strategies, directed toward ensuring an 'ethical' device, would seem requisite to the sucessful application of any form of hyperconscious machine technology

ON SPECTRUM OF I-GRAPHS AND ITS ORDERING WITH RESPECT TO SPECTRAL MOMENTS

Suppose $G$ is a graph, $A(G)$ its adjacency matrix, and $μ_1(G), μ_2(G), \cdots, μ_n(G)$ are eigenvalues of $A(G)$. The numbers $S_k(G) = \sum_{i=1}^n μ^k_i(G)$, $0 \leq k \leq n − 1$, are said to be the k−th spectral moment of $G$ and the sequenceS(G) = (S_0(G), S_1(G), \sdots, S_{n−1}(G)) is called the spectral moments sequence of $G$. For two graphs $G_1$ and $G_2$, we define $G_1 \leq_S G_2$, if there exists an integer$k$, $1 \leq k \leq n − 1$, such that for each $i$, $0 \leq i \leq k − 1$, $S_i(G_1) = S_i(G_2)$ andS_k(G_1) < S_k(G_2).The I−graph $I(n, j, k)$ is a graph of order $2n$ with the vertex and edge sets$V(I(n, j, k) = \{u_0, u_1, \cdots, u_{n−1}, v_0, v_1, \cdots, v_{n−1}\}$,$E(I(n, j, k) = \{u_iu{u+j}, u_iv_i, v_iv_{i+k} ; 0 \leq i \leq n − 1\}$,respectively. The aim of this paper is to compute the spectrum of an arbitraryI−graph and the extremal I−graphs with respect to the S−order