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In this note we consider the independent domination polynomial problem along with their unimodal and log-concave properties which were earlier studied by G\"ursoy, \"Ulker and G\"ursoy (Soft Comp. 2022). We show that the independent domination polynomial of zero divisor graphs of $\mathbb{Z}_{n}$ for $n\in \{ pq, p^{2}q, pqr, p^{\alpha}\}$ where $p,q,r$ are primes with $2

For a commutative ring $R,$ with non-zero zero divisors $Z^{\ast}(R)$. The zero divisor graph $\Gamma(R)$ is a simple graph with vertex set $Z^{\ast}(R)$, and two distinct vertices $x,y\in V(\Gamma(R))$ are adjacent if and only if $x\cdot y=0.$ In this note, we provide counter examples to the eigenvalues, the energy and the second Zagreb index related to zero divisor graphs of rings obtained in [Johnson and Sankar, J. Appl. Math. Comp. (2023), \cite{johnson}]. We correct the eigenvalues (energy) and the Zagreb index result for the zero divisor graphs of ring $\mathbb{Z}_{p}[x]/\langle x^{4} \rangle.$ We show that for any prime $p$, $\Gamma(\mathbb{Z}_{p}[x]/\langle x^{4} \rangle)$ is non-hyperenergetic and for prime $p\geq 3$, $\Gamma(\mathbb{Z}_{p}[x]/\langle x^{4} \rangle)$ is hypoenergetic. We give a formulae for the topological indices of $\Gamma(\mathbb{Z}_{p}[x]/\langle x^{4} \rangle)$ and show that its Zagreb indices satisfy Hansen and Vuki$\check{c}$cevi\'c conjecture \cite{hansen}., Comment: 20 pages, 3 Figures, Submitted to journal "Journal of Applied Mathematics and Computing" on 10 Apr 2023, Comments and suggestions are welcome and can be sent at bilalahmadrr@gamil.com

The proximity $\pi = \pi (G)$ of a connected graph $G$ is the minimum, over all vertices, of the average distance from a vertex to all others. Similarly, the maximum is called the remoteness and denoted by $\rho = \rho (G)$. The concepts of proximity and remoteness, first defined in 2006, attracted the attention of several researchers in Graph Theory. Their investigation led to a considerable number of publications. In this paper, we present a survey of the research work., Comment: arXiv admin note: substantial text overlap with arXiv:1204.1184 by other authors, updated version, to appear in Discrete Applied Mathematics

For a simple connected graph $G$ of order $n$ having distance Laplacian eigenvalues $ \rho^{L}_{1}\geq \rho^{L}_{2}\geq \cdots \geq \rho^{L}_{n}$, the distance Laplacian energy $DLE(G)$ is defined as $DLE(G)=\sum_{i=1}^{n}\left|\rho^{L}_i-\frac{2 W(G)}{n}\right|$, where $W(G)$ is the Wiener index of $G$. We obtain a relationship between the Laplacian energy and distance Laplacian energy for graphs with diameter 2. We obtain lower bounds for the distance Laplacian energy $DLE(G)$ in terms of the order $n$, the Wiener index $W(G)$, independence number, vertex connectivity number and other given parameters. We characterize the extremal graphs attaining these bounds. We show that the complete bipartite graph has the minimum distance Laplacian energy among all connected bipartite graphs and complete split graph has the minimum distance Laplacian energy among all connected graphs with given independence number. Further, we obtain the distance Laplacian spectrum of the join of a graph with the union of two other graphs. We show that the graph $K_{k}\bigtriangledown(K_{t}\cup K_{n-k-t}), 1\leq t \leq \lfloor\frac{n-k}{2}\rfloor $, has the minimum distance Laplacian energy among all connected graphs with vertex connectivity $k$. We conclude this paper with a discussion on trace norm of a matrix and the importance of our results in the theory of trace norm of the matrix $D^L(G)-\frac{2W(G)}{n}I_n$., Comment: The paper is submitted for possible publication in the Czechoslovak Mathematical Journal, Sep 2020

For a simple graph $G$, the generalized adjacency matrix $A_{\alpha}(G)$ is defined as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G), \alpha\in [0,1]$, where $A(G)$ is the adjacency matrix and $D(G)$ is the diagonal matrix of vertex degrees of $G$. This matrix generalises the spectral theories of the adjacency matrix and the signless Laplacian matrix of $G$. In this paper, we find $ A_{\alpha} $-spectrum of the joined union of graphs in terms of spectrum of adjacency matrices of its components and the zeros of the characteristic polynomials of an auxiliary matrix determined by the joined union. We determine the $ A_{\alpha}$-spectrum of join of two regular graphs, the join of a regular graph with the union of two regular graphs of distinct degrees. As an applications, we investigate the $ A_{\alpha} $-spectrum of certain power graphs of finite groups., Comment: Submitted to Journal of Algebra and its Applications on August 2020

Let $G$ be a simple graph with $n$ vertices, $m$ edges having Laplacian eigenvalues $\mu_1, \mu_2, \dots, \mu_{n-1},\mu_n=0$. The Laplacian energy $LE(G)$ is defined as $LE(G)=\sum_{i=1}^{n}|\mu_i-\overline{d}|$, where $\overline{d}=\frac{2m}{n}$ is the average degree of $G$. Radenkovi\'{c} and Gutman conjectured that among all trees of order $n$, the path graph $P_n$ has the smallest Laplacian energy. Let $ \mathcal{T}_{n}(d) $ be the family of trees of order $n$ having diameter $ d $. In this paper, we show that Laplacian energy of any tree $T\in \mathcal{T}_{n}(4)$ is greater than the Laplacian energy of $P_n$, thereby proving the conjecture for all trees of diameter $4$. We also show the truth of conjecture for all trees with number of non-pendent vertices at most $\frac{9n}{25}-2$. Further, we give some sufficient conditions for the conjecture to hold for a tree of order $n$.

The work in this thesis concerns the investigation of eigenvalues of the Laplacian matrix, normalized Laplacian matrix, signless Laplacian matrix and distance signless Laplacian matrix of graphs. In Chapter 1, we present a brief introduction of spectral graph theory with some definitions. Chapter $2$ deals with the sum of $ k $ largest Laplacian eigenvalues $ S_{k}(G) $ of graph $ G $ and Brouwer's conjecture. We obtain the upper bounds for $ S_{k}(G) $ for some classes of graphs and use them to verify Brouwer's conjecture for these classes of graphs. Also, we prove Brouwer's conjecture for more general classes of graphs. In Chapter $3$, we investigate the Laplacian eigenvalues of graphs and the Laplacian energy conjecture for trees. We prove the Laplacian energy conjecture completely for trees of diameter $ 4 $. Further, we prove this conjecture for all trees having at most $ \frac{9n}{25}-2 $ non-pendent vertices. Also, we obtain the sufficient conditions for the truth of conjecture for trees of order $ n $. In Chapter $4$, we determine the normalized Laplacian spectrum of the joined union of regular graphs and obtain the spectrum of some well known graphs. As consequences of joined union, we obtain the normalized Laplacian spectrum of power graphs associated to finite cyclic groups. In Chapter $5$, we find the distance signless Laplacian spectrum of regular graphs and zero-divisor graphs associated to finite commutative ring. Also, we find the bounds for spectral radius of generalized distance matrix. Further, we obtain the generalized distance energy for bipartite graphs and trees. We prove that the complete bipartite graph has minimum generalized distance energy among all connected bipartite graphs. Besides, for $ \alpha\in \big(0, \frac{2n}{3n-2}\big) $, we show that the star graph has minimum generalized distance energy among all trees., Comment: P.hD Thesis submitted in mathematics at University of Kashmir, Srinagar Kashmir 190006, India. The thesis is also available at http://hdl.handle.net/10603/329592

For a simple connected graph $ G $ of order $ n $, the normalized Laplacian is a square matrix of order $ n $, defined as $\mathcal{L}(G)= D(G)^{-\frac{1}{2}}L(G)D(G)^{-\frac{1}{2}}$, where $ D(G)^{-\frac{1}{2}} $ is the diagonal matrix whose $ i$-th diagonal entry is $ \frac{1}{\sqrt{d_{i}}} $. In this article, we find the normalized Laplacian eigenvalues of the joined union of regular graphs in terms of the adjacency eigenvalues and the eigenvalues of quotient matrix associated with graph $ G $. For a finite group $\mathcal{G}$, the power graph $\mathcal{P}(\mathcal{G})$ of a group $ \mathcal{G} $ is defined as the simple graph in which two distinct vertices are joined by an edge if and only if one is the power of other. As a consequence of the joined union of graphs, we investigate the normalized Laplacian eigenvalues of power graphs of finite cyclic group $ \mathbb{Z}_{n}. $, Comment: 23 pages

Let $A(G)$ be the adjacency matrix and $D(G)$ be the diagonal matrix of the vertex degrees of a simple connected graph $G$. Nikiforov defined the matrix $A_{\alpha}(G)$ of the convex combinations of $D(G)$ and $A(G)$ as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, for $0\leq \alpha\leq 1$. If $ \rho_{1}\geq \rho_{2}\geq \dots \geq \rho_{n}$ are the eigenvalues of $A_{\alpha}(G)$ (which we call $\alpha$-adjacency eigenvalues of $G$), the $ \alpha $-adjacency energy of $G$ is defined as $E^{A_{\alpha}}(G)=\sum_{i=1}^{n}\left|\rho_i-\frac{2\alpha m}{n}\right|$, where $n$ is the order and $m$ is the size of $G$. We obtain the upper and lower bounds for $E^{A_{\alpha}}(G) $ in terms of order $n$, size $m$ and Zagreb index $Zg(G)$ associated to the structure of $G$. Further, we characterize the extremal graphs attaining these bounds., Comment: 17 pages