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In this paper, image fusion method based on a new class of wavelet — non-separable wavelet with compactly supported, linear phase, orthogonal and dilation matrix is presented. We first construct a non-separable wavelet filter bank. Using these filters, the images involved are decomposed into wavelet pyramids. Then the following fusion algorithm was proposed: for low-frequency part, the average value is selected for new pixel value, For the three high-frequency parts of each level, the standard deviation of each image patch over 3×3 window in the high-frequency sub-images is computed as activity measurement. If the standard deviation of the area 3×3 window is bigger than the standard deviation of the corresponding 3×3 window in the other high-frequency sub-image. The center pixel values of the area window that the weighted area energy is bigger are selected. Otherwise the weighted value of the pixel is computed. Then a new fused image is reconstructed. The performance of the method is evaluated using the entropy, cross-entropy, fusion symmetry, root mean square error and peak-to-peak signal-to-noise ratio. The experiment results show that the non-separable wavelet fusion method proposed in this paper is very close to the performance of the Haar separable wavelet fusion method.

The energy of a weighted digraph $D$ is the sum of absolute values of real part of its eigenvalues. Recently, the minimal and maximal energy of unicyclic weighted digraphs with cycle weight $r\in [-1,1]\setminus \{0\}$ is studied. In this paper, we introduce a class $𝔹(r,s)$ of those bicyclic weighted digraphs of fixed order which contain vertex-disjoint weighted cycles of weights $r$ or $-r$ and $s$ or $-s$, where $0<r,s\le 1$. We find digraphs in $𝔹(r,s)$ under certain conditions on $r$ and $s$ with extremal energy.

The energy of a square matrix is the sum of the absolute values of deviations of its eigenvalues from their mean. In this paper, we extend the notion of the Laplacian and the signless Laplacian energy of a graph to non-negative real symmetric matrices with zero diagonal. Such a matrix $\widehat{A}$ can be considered as the adjacency matrix of some weighted graph. Considering the sum of the weights of edges that are incident to a vertex as the weight of the corresponding vertex, we construct the diagonal matrix $\widehat{D}$ of which the diagonal entries are the weights of the corresponding vertices. We define a weighted version of the Laplacian matrix as $\widehat{L}=\widehat{D}-\widehat{A}$ and that of the signless Laplacian matrix as $\widehat{Q}=\widehat{D}+\widehat{A}$. In this work, we obtain some bounds of the energies of the weighted adjacency matrix, the weighted Laplacian matrix, and the weighted signless Laplacian matrix by studying the corresponding weighted graph. We also obtain some relations interconnecting those energies.

A mixed graph is a graph whose edge set consists of both oriented and unoriented edges. The Hermitian-adjacency matrix of an $n$-vertex mixed graph is a square matrix $H(M)=[h_{jk}]$ of order $n$, where $h_{jk}=\dot{\iota }=-h_{kj}$ if there is an arc from $v_j$ to $v_k$ and $h_{jk}=1$ if there is an edge between $v_j$ and $v_k$, and $h_{jk}=0$ otherwise. Let $D(M)=[d_{jj}]$ be a diagonal matrix, where $d_{jj}$ is the degree of $v_j$ in the underlying graph of $M$. The matrices $L(M)=D(M)-H(M)$ and $Q(M)=D(M)+H(M)$ are, respectively, the Hermitian Laplacian and Hermitian quasi-Laplacian matrix of the mixed graph $M$. In this paper, we first found coefficients of the characteristic polynomial of Hermitian Laplacian and Hermitian quasi-Laplacian matrices of the mixed graph $M$. Second, we discussed relationship between the spectra of Hermitian Laplacian and Hermitian quasi-Laplacian matrices of the mixed graph $M$.

Let ${ℛ}$ be commutative ring and $Z^{\ast }({ℛ})$ be the set of all non-zero zero divisors of ${ℛ}$. Then $\mathrm{\Gamma }({ℛ})$ is said to be the zero divisor graph if and only if $a\cdot b=0$ where $a,b\in V(\mathrm{\Gamma }({ℛ}))$ and $(a,b)\in E(\mathrm{\Gamma }({ℛ}))$. Graph energy ${ℰ}(\mathrm{\Gamma }({ℛ}))$ is defined as the sum of the absolute eigenvalues of the adjacency matrix ${ℳ}(\mathrm{\Gamma }({ℛ}))$, then ${ℰ}(\mathrm{\Gamma }({ℛ}))={\sum }_{i=1}^n\left|{\lambda }_i\right|$. Wiener index $W(\mathrm{\Gamma }({ℛ}))$ is defined as the sum of all distance between pairs of vertices $a$ and $b$, then $W(\mathrm{\Gamma }({ℛ}))={\sum }_{a,b\in V(\mathrm{\Gamma }({ℛ}))}d(a,b)$. In this paper, we compute the graph energy from the adjacency matrix of the zero divisor graph and the Wiener index from the zero divisor graph associated with commutative rings.