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Neural networks have emerged as powerful tools across various applications, yet their decision-making process often remains opaque, leading to them being perceived as "black boxes." This opacity raises concerns about their interpretability and reliability, especially in safety-critical scenarios. Network inversion techniques offer a solution by allowing us to peek inside these black boxes, revealing the features and patterns learned by the networks behind their decision-making processes and thereby provide valuable insights into how neural networks arrive at their conclusions, making them more interpretable and trustworthy. This paper presents a simple yet effective approach to network inversion using a carefully conditioned generator that learns the data distribution in the input space of the trained neural network, enabling the reconstruction of inputs that would most likely lead to the desired outputs. To capture the diversity in the input space for a given output, instead of simply revealing the conditioning labels to the generator, we hideously encode the conditioning label information into vectors, further exemplified by heavy dropout in the generation process and minimisation of cosine similarity between the features corresponding to the generated images. The paper concludes with immediate applications of Network Inversion including in interpretability, explainability and generation of adversarial samples.

In this article, we explore the concept of spectral redundancy within the class of pineapple graphs, denoted as $\mathcal{P}(\alpha,\beta)$. These graphs are constructed by attaching $\beta$ pendent edges to a single vertex of a complete graph $K_\alpha$. A connected graph $G$ earns the title of being spectrally non-redundant if the spectral radii of its connected induced subgraphs remain distinct. Spectral redundancy, on the other hand, arises when there is a repetition of spectral radii among the connected induced subgraphs within $G$. Specifically, we analyze the adjacency spectrum of $\mathcal{P}(\alpha,\beta)$, revealing distinct eigenvalues including $0$, $-1$, and additional eigenvalues, some negative and others positive. Our investigation focuses on determining the spectral redundancy within this class of graphs, shedding light on their unique structural properties and implications for graph theory., Comment: 16 pages

While the deployment of neural networks, yielding impressive results, becomes more prevalent in various applications, their interpretability and understanding remain a critical challenge. Network inversion, a technique that aims to reconstruct the input space from the model's learned internal representations, plays a pivotal role in unraveling the black-box nature of input to output mappings in neural networks. In safety-critical scenarios, where model outputs may influence pivotal decisions, the integrity of the corresponding input space is paramount, necessitating the elimination of any extraneous "garbage" to ensure the trustworthiness of the network. Binarised Neural Networks (BNNs), characterized by binary weights and activations, offer computational efficiency and reduced memory requirements, making them suitable for resource-constrained environments. This paper introduces a novel approach to invert a trained BNN by encoding it into a CNF formula that captures the network's structure, allowing for both inference and inversion.

Let $G$ be a simple connected graph of size $m$. Let $A$ be the adjacency matrix of $G$ and let $\rho(G)$ be the spectral radius of $G$. A graph is said to be $H$-free if it does not contain a subgraph isomorphic to $H$. Let $H(\ell,3)$ be the graph formed by taking a cycle of length $\ell$ and a triangle on a common vertex. Recently, Li, Lu and Peng [Y. Li, L. Lu, Y. Peng, Spectral extremal graphs for the bowtie, Discrete Math. 346(12) (2023) 113680.] showed that the unique $m$-edge $H(3,3)$-free spectral extremal graph is the join of $K_2$ with an independent set of $\frac{m-1}{2}$ vertices if $m\ge 8$ and the condition $m\ge 8$ is tight. In particular, if $G$ does not contain $H(3,3)$ as induced subgraph, they proved that $\rho(G) \leq \frac{1+\sqrt{4m-3}}{2} $ and equality holds when $G$ is isomorphic to $S_{\frac{m+3}{2},2}$. Note that Li et al. denoted $H(3,3)$ by $F_2$. In this paper, we find the maximum spectral radius and identify the graph with the largest spectral radius among all \{$H(3,3), H(4,3)$\}-free graphs of size odd $m$, where $m\geq 259$. Coincidentally, we show that $\rho(G) \leq \frac{1+\sqrt{4m-3}}{2}$ when $G$ forbids both $H(3,3)$ and $H(4,3)$. In our case, the equality holds when $G$ is isomorphic to the same graph., Comment: 15 pages

In this paper, we introduce a new graph structure, called the $direct~ sum ~graph$ on a finite dimensional vector space. We investigate the connectivity, diameter and the completeness of $\Gamma_{U\oplus W}(\mathbb{V})$. Further, we find its domination number and independence number. We also determine the degree of each vertex in case the base field is finite and show that the graph $\Gamma_{U\oplus W}(\mathbb{V})$ is not Eulerian. We also show that under some mild conditions the graph $\Gamma_{U\oplus W}(\mathbb{V})$ is triangulated. We determine the clique number of $\Gamma_{U\oplus W}(\mathbb{V})$ for some particular cases. Finally, we find the size, girth, edge-connectivity and the chromatic number of $\Gamma_{U\oplus W}(\mathbb{V})$., Comment: 19 pages

Let $\Gamma=(G, \sigma)$ be a signed graph of order $n$ with eigenvalues $\mu_1,\mu_2,\ldots,\mu_n.$ We define the Estrada index of a signed graph $\Gamma$ as $EE(\Gamma)=\sum_{i=1}^ne^{\mu_i}$. We characterize the signed unicyclic graphs with the maximum Estrada index. The signed graph $\Gamma$ is said to have the pairing property if $\mu$ is an eigenvalue whenever $-\mu$ is an eigenvalue of $\Gamma$ and both $\mu$ and $-\mu$ have the same multiplicities. If $\Gamma_{p}^-(n, m)$ denotes the set of all unbalanced graphs on $n$ vertices and $m$ edges with the pairing property, we determine the signed graphs having the maximum Estrada index in $\Gamma_{p}^-(n, m)$, when $m=n$ and $m=n+1$. Finally, we find the signed graphs among all unbalanced complete bipartite signed graphs having the maximum Estrada index., Comment: 16 pages, 2 figures

A signed graph $S=(G, \sigma)$ is a pair in which $G$ is an underlying graph and $\sigma$ is a function from the edge set to $\{\pm1\}$. For signed graphs $S_{1}$ and $S_{2}$ on $n_{1}$ and $n_{2}$ vertices, respectively, the signed neighbourhood corona $S_{1} \star_s S_{2}$ (in short s-neighbourhood corona) of $S_{1}$ and $S_{2}$ is the signed graph obtained by taking one copy of $S_{1}$ and $n_{1}$ copies of $S_{2}$ and joining every neighbour of the $i$th vertex of $S_{1}$ with the same sign as the sign of incident edge to every vertex in the $i$th copy of $S_{2}$. In this paper, we investigate the adjacency, Laplacian and net Laplacian spectrum of $S_{1} \star_s S_{2}$ in terms of the corresponding spectrum of $ S_{1}$ and $ S_{2}$. We determine $(i)$ the adjacency spectrum of $S_{1} \star_s S_{2}$ for arbitrary $S_{1} $ and net regular $ S_{2}$, $(ii)$ the Laplacian spectrum for regular $S_{1} $ and regular and net regular $ S_{2}$ and $(iii)$ the net Laplacian spectrum for net regular $S_{1} $ and arbitrary $ S_{2}$. As a consequence, we obtain the signed graphs with $4$ and $5$ distinct adjacency, Laplacian and net Laplacian eigenvalues. Finally, we show that the signed neighbourhood corona of two signed graphs is not determined by its adjacency (resp., Laplacian, net Laplacian) spectrum., Comment: 16 pages

For a graph $G$, the generalized adjacency matrix $A_\alpha(G)$ is the convex combination of the diagonal matrix $D(G)$ and the adjacency matrix $A(G)$ and is defined as $A_\alpha(G)=\alpha D(G)+(1-\alpha) A(G)$ for $0\leq \alpha \leq 1$. This matrix has been found to be useful in merging the spectral theories of $A(G)$ and the signless Laplacian matrix $Q(G)$ of the graph $G$. The generalized adjacency energy or $A_\alpha$-energy is the mean deviation of the $A_\alpha$-eigenvalues of $G$ and is defined as $E(A_\alpha(G))=\sum_{i=1}^{n}|p_i-\frac{2\alpha m}{n}|$, where $p_i$'s are $A_\alpha$-eigenvalues of $G$. In this paper, we investigate the $A_\alpha$-eigenvalues of a strongly regular graph $G$. We observe that $A_\alpha$-spectral radius $p_1$ satisfies $\delta(G)\leq p_1 \leq \Delta(G)$, where $\delta(G)$ and $\Delta(G)$ are, respectively, the smallest and the largest degrees of $G$. Further, we show that the complete graph is the only graph to have exactly two distinct $A_\alpha$-eigenvalues. We obtain lower and upper bounds of $A_\alpha$-energy in terms of order, size and extremal degrees of $G$. We also discuss the extremal cases of these bounds.

Plastic inhomogeneity, particularly localised strain, is one of the main mechanisms responsible for failures in engineering alloys. This work studied the spatial arrangement and distribution of microstructure and their influence in the plastic inhomogeneity of Inconel 718 fabricated by additive manufacturing (AM). In particular, the density and spatial distribution of geometrically necessary dislocations (GNDs) and the crystallographic texture components were examined in relation to the spatial plastic deformation. In-situ tensile testing coupled with microscopy and EBSD was performed to understand the evolution of the as-built microstructure and its effect on the spatial and temporal deformation behaviour. Bidirectional scan strategy with no interlayer rotation resulted in highly ordered alternating arrangements of coarse Goss-like {110}<001> textured grains separated by fine Cube-like {100}<001> textured grains. This strategy also resulted in an overall high density of dislocations (in particular GNDs) with a regular pattern: the fine grains contained very high GND density while the coarse grains contained less GNDs but they were oriented to be perpendicular to the scan direction. Although this strategy resulted in a dominant Cube texture that is desirable for isotropy, the Cube texture gradually weakened and the undesirable Goss component increased during plastic deformation. Together with the columnar grain morphology and aligned GND arrangement, such texture evolution caused intense plastic inhomogeneity during deformation. Replacing the bidirectional strategy with a chessboard strategy involving a 67{\deg} layer rotation resulted in a lower GND density and much more random distribution of crystallographic texture and GNDs, with a dominant Cube component (and lower Goss texture) that remained stable throughout plastic deformation, reducing the plastic inhomogeneity.

Let $G$ be a simple connected simple graph of order $n$. The distance Laplacian matrix $D^{L}(G)$ is defined as $D^L(G)=Diag(Tr)-D(G)$, where $Diag(Tr)$ is the diagonal matrix of vertex transmissions and $D(G)$ is the distance matrix of $G$. The eigenvalues of $D^{L}(G)$ are the distance Laplacian eigenvalues of $G$ and are denoted by $\partial_{1}^{L}(G), \partial_{2}^{L}(G),\dots,\partial_{n}^{L}(G)$. The \textit{ distance Laplacian spread} $DLS(G)$ of a connected graph $G$ is the difference between largest and second smallest distance Laplacian eigenvalues, that is, $\partial_{1}^{L}(G)-\partial_{n-1}^{L}(G)$. We obtain bounds for $DLS(G)$ in terms of the Wiener index $W(G)$, order $n$ and the maximum transmission degree $Tr_{max}(G)$ of $G$ and characterize the extremal graphs. We obtain two lower bounds for $DLS(G)$, the first one in terms of the order, diameter and the Wiener index of the graph, and the second one in terms of the order, maximum degree and the independence number of the graph. For a connected $ k-partite$ graph $G$, $k\leq n-1$, with $n$ vertices having disconnected complement, we show that $ DLS(G)\geq \Big \lfloor \frac{n}{k}\Big \rfloor$ with equality if and only if $G$ is a $complete ~ k-partite$ graph having cardinality of each independent class same and $n \equiv 0 \pmod k$., Comment: 14 pages

The reciprocal distance Laplacian matrix of a connected graph $G$ is defined as $RD^L(G)=RT(G)-RD(G)$, where $RT(G)$ is the diagonal matrix of reciprocal distance degrees and $RD(G)$ is the Harary matrix. Since $RD^L(G)$ is a real symmetric matrix, we denote its eigenvalues as $\lambda_1(RD^L(G))\geq \lambda_2(RD^L(G))\geq \dots \geq \lambda_n(RD^L(G))$. The largest eigenvalue $\lambda_1(RD^L(G))$ of $RD^L(G)$ is called the reciprocal distance Laplacian spectral radius. In this article, we prove that the multiplicity of $n$ as a reciprocal distance Laplacian eigenvalue of $RD^L(G)$ is exactly one less than the number of components in the complement graph $\bar{G}$ of $G$. We show that the class of the complete bipartite graphs maximize the reciprocal distance Laplacian spectral radius among all the bipartite graphs with $n$ vertices. Also, we show that the star graph $S_n$ is the unique graph having the maximum reciprocal distance Laplacian spectral radius in the class of trees with $n$ vertices. We determine the reciprocal distance Laplacian spectrum of several well known graphs. We prove that the complete graph $K_n$, $K_n-e$, the star $S_n$, the complete balanced bipartite graph $K_{\frac{n}{2},\frac{n}{2}}$ and the complete split graph $CS(n,\alpha)$ are all determined from the $RD^L$-spectrum., Comment: 13 pages

Let $\Gamma=(G,\sigma)$ be a signed graph, where $\sigma$ is the sign function on the edges of $G$. In this paper, we use the operation of partial transpose to obtain non-isomorphic Laplacian cospectral signed graphs. We will introduce two new operations on signed graphs. These operations will establish a relationship between the adjacency spectrum of one signed graph with the Laplacian spectrum of another signed graph. As an application, these new operations will be utilized to construct several pairs of cospectral non-isomorphic signed graphs. Finally, we construct integral signed graphs., Comment: 23 pages, 10 figures

Let $G$ be a simple graph with order $n$ and size $m$. The quantity $M_1(G)=\displaystyle\sum_{i=1}^{n}d^2_{v_i}$ is called the first Zagreb index of $G$, where $d_{v_i}$ is the degree of vertex $v_i$, for all $i=1,2,\dots,n$. The signless Laplacian matrix of a graph $G$ is $Q(G)=D(G)+A(G)$, where $A(G)$ and $D(G)$ denote, respectively, the adjacency and the diagonal matrix of the vertex degrees of $G$. Let $q_1\geq q_2\geq \dots\geq q_n\geq 0$ be the signless Laplacian eigenvalues of $G$. The largest signless Laplacian eigenvalue $q_1$ is called the signless Laplacian spectral radius or $Q$-index of $G$ and is denoted by $q(G)$. Let $S^+_k(G)=\displaystyle\sum_{i=1}^{k}q_i$ and $L_k(G)=\displaystyle\sum_{i=0}^{k-1}q_{n-i}$, where $1\leq k\leq n$, respectively denote the sum of $k$ largest and smallest signless Laplacian eigenvalues of $G$. The signless Laplacian energy of $G$ is defined as $QE(G)=\displaystyle\sum_{i=1}^{n}|q_i-\overline{d}|$, where $\overline{d}=\frac{2m}{n}$ is the average vertex degree of $G$. In this article, we obtain upper bounds for the first Zagreb index $M_1(G)$ and show that each bound is best possible. Using these bounds, we obtain several upper bounds for the graph invariant $S^+_k(G)$ and characterize the extremal cases. As a consequence, we find upper bounds for the $Q$-index and lower bounds for the graph invariant $L_k(G)$ in terms of various graph parameters and determine the extremal cases. As an application, we obtain upper bounds for the signless Laplacian energy of a graph and characterize the extremal cases., Comment: 13

Let $R$ be a commutative ring with unity, $M$ be a unitary $R$-module and $G$ a finite abelian group (viewed as a $\mathbb{Z}$-module). The main objective of this paper is to study properties of mod-annihilators of $M$. For $x \in M$, we study the ideals $[x : M] =\{r\in R | rM\subseteq Rx\}$ of $R$ corresponding to mod-annihilator of $M$. We investigate that when $[x : M]$ is an essential ideal of $R$. We prove that arbitrary intersection of essential ideals represented by mod-annihilators is an essential ideal. We observe that $[x : M]$ is injective if and only if $R$ is non-singular and the radical of $R/[x : M]$ is zero. Moreover, if essential socle of $M$ is non-zero, then we show that $[x : M]$ is the intersection of maximal ideals and $[x : M]^2 = [x : M]$. Finally, we discuss the correspondence of essential ideals of $R$ and vertices of the annihilating graphs realized by $M$ over $R$., Comment: arXiv admin note: substantial text overlap with arXiv:1711.01130

Let $G$ be a connected simple graph with $n$ vertices. The distance Laplacian matrix $D^{L}(G)$ is defined as $D^L(G)=Diag(Tr)-D(G)$, where $Diag(Tr)$ is the diagonal matrix of vertex transmissions and $D(G)$ is the distance matrix of $G$. The eigenvalues of $D^{L}(G)$ are the distance Laplacian eigenvalues of $G$ and are denoted by $\partial_{1}^{L}(G)\geq \partial_{2}^{L}(G)\geq \dots \geq \partial_{n}^{L}(G)$. The largest eigenvalue $\partial_{1}^{L}(G)$ is called the distance Laplacian spectral radius. Lu et al. (2017), Fernandes et al. (2018) and Ma et al. (2018) completely characterized the graphs having some distance Laplacian eigenvalue of multiplicity $n-3$. In this paper, we characterize the graphs having distance Laplacian spectral radius of multiplicity $n-4$ together with one of the distance Laplacian eigenvalue as $n$ of multiplicity either 3 or 2. Further, we completely determine the graphs for which the distance Laplacian eigenvalue $n$ is of multiplicity $n-4$., Comment: 11 pages

For a connected graph $G$ of order $n$, let $Diag(Tr)$ be the diagonal matrix of vertex transmissions and $D(G)$ be the distance matrix of $G$. The distance Laplacian matrix of $G$ is defined as $D^L(G)=Diag(Tr)-D(G)$ and the eigenvalues of $D^{L}(G)$ are called the distance Laplacian eigenvalues of $G$. Let $\partial_{1}^{L}(G)\geq \partial_{2}^{L}(G)\geq \dots \geq \partial_{n}^{L}(G)$ be the distance Laplacian eigenvalues of $G$. Given an interval $I$, let $m_{D^{L} (G)} I$ (or simply $m_{D^{L} } I$) be the number of distance Laplacian eigenvalues of $G$ which lie in the interval $I$. For a prescribed interval $I$, we determine $m_{D^{L} }I$ in terms of independence number $\alpha(G)$, chromatic number $\chi$, number of pendant vertices and diameter $d$ of the graph $G$. In particular, we prove that $m_{D^{L}(G) }[n,n+2)\leq \chi-1$, ~$m_{D^{L}(G) }[n,n+\alpha(G))\leq n-\alpha(G)$ and we show that the inequalities are sharp. We also show that $m_{D^{L} (G )}\bigg( n,n+\left\lceil\frac{n}{\chi}\right\rceil\bigg)\leq n- \left\lceil\frac{n}{\chi}\right\rceil-C_{\overline{G}}+1 $, where $C_{\overline{G}}$ is the number of components in $\overline{G}$, and discuss some cases where the bound is best possible. In addition, we prove that $m_{D^{L} (G )}[n,n+p)\leq n-p$, where $p\geq 1$ is the number of pendant vertices. Also, we characterize graphs of diameter $d\leq 2$ which satisfy $m_{D^{L}(G) } (2n-1,2n )= \alpha(G)-1=\frac{n}{2}-1$. At the end, we propose some problems of interest., Comment: 17 pages