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Soft set theory, proposed by D. Molodtsov, is a mathematical framework for dealing with uncertain data. Soft set theory is now widely used to solve decision-making problems and has a wide range of applications in economics, engineering, medicine, and other fields, as demonstrated by Molodtsov’s pioneering work. A directed graph is a graph with directed edges. Directed graphs can be used to study and solve problems involving social networks, shortest paths, electrical circuits, and so on. By extending the notion of the soft set to directed graphs, we presented soft directed graphs. Soft directed graphs offer a parameterized perspective on directed graphs. In this paper, we introduce and investigate the corona product and the restricted corona product of soft directed graphs.

The concept of zero-divisor graph of a commutative ring was introduced by Beck [Coloring of commutating ring, J. Algebra116 (1988) 208–226]. In this paper, we present some properties of zero divisor graphs obtained from ring $Z_p\times Z_q\times Z_r$, where $p,q$ and $r$ are primes. Also, we give some degree-based topological indices of this special graph.

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. The general reduced second Zagreb index of $G$ is defined as ${\mathrm{\text{GRM}}}_{\beta }(G)={\sum }_{uv\in E(G)}(d(u)+\beta )(d(v)+\beta )$, where $\beta$ is any real number and $d(v)$ is the degree of the vertex $v$ of $G$. In this paper, the general reduced second Zagreb index of the Cartesian product, corona product, join of graphs and two new operations of graphs are computed.

The distance, distance signless Laplacian and distance Laplacian matrix of a simple connected graph $G,$ are denoted by $D(G),D_Q(G)=D(G)+\mathrm{\text{Tr}}(G)$ and $D_L(G)=\mathrm{\text{Tr}}(G)-D(G),$ respectively, where $\mathrm{\text{Tr}}(G)$ is the diagonal matrix of vertex transmission. Ger$\check{\mathrm{\text{s}}}$gorin discs for any $n\times n$ square matrix $A=[a_{ij}]$ are the discs $\{z\in ℂ:|z-a_{ii}|\le R_i(A)\}$, where $R_i(A)={\sum }_{j\ne i}\left|a_{ij}\right|,i=1,2,\dots ,n$. The famous Ger$\check{\mathrm{\text{s}}}$gorin disc theorem says that all the eigenvalues of a square matrix lie in the union of the Ger$\check{\mathrm{\text{s}}}$gorin discs of that matrix. In this paper, some classes of graphs are studied for which the smallest Ger$\check{\mathrm{\text{s}}}$gorin disc contains every distance and distance signless Laplacian eigenvalues except the spectral radius of the corresponding matrix. For all connected graphs, a lower bound and for trees, an upper bound of every distance signless Laplacian eigenvalues except the spectral radius is given in this paper. These bounds are comparatively better than the existing bounds. By applying these bounds, we find some infinite classes of graphs for which the smallest Ger$\check{\mathrm{\text{s}}}$gorin disc contains every distance signless Laplacian eigenvalues except the spectral radius of the distance signless Laplacian matrix. For the distance Laplacian eigenvalues, we have given an upper bound and then find a condition for which the smallest Ger$\check{\mathrm{\text{s}}}$gorin disc contains every distance Laplacian eigenvalue of the distance Laplacian matrix. These results give partial answers from some questions that are raised in [2].

In this paper, we investigate when the generalized join product, generalized corona product, generalized edge corona product and the Kronecker product of graphs are divisor graphs. Also the situations under which the zero divisor graph and the co-maximal graph are divisor graphs are characterized.

In this paper, we present exact formula for the weighted PI index of corona product of two connected graphs in terms of other graph invariants including the PI index, first Zagreb index and second Zagreb index. Then, we apply our result to compute the weighted PI indices of t-fold bristled graph, bottleneck graph, sunlet graph, star graph, fan graph, wheel graph and some classes of bridge graphs.

A set S of vertices in a connected graph $G=(V,E)$ is called a geodeticset if every vertex not in $S$ lies on a shortest path between two vertices from $S$. A set $D$ of vertices in $G$ is called a dominating set of $G$ if every vertex not in $D$ has at least one neighbor in $D$. A set $S$ is called a geodetic global dominating set of $G$ if $S$ is both geodetic and global dominating set of $G$. The geodetic global domination number is the minimum cardinality of a geodetic global dominating set in $G$. In this paper, we determine the geodetic global domination number of the corona and strong products of two graphs.

Let $G$ be a simple graph with vertex set $V(G)$ and edge set $E(G)$, respectively. An edge irregular $k$-labeling of $G$ is a labeling of $V(G)$ with labels from the set $\{1,2,\dots ,k\}$ in such a way that for any two different edges $e$ and $f$, their weights $w(e)$ and $w(f)$ are distinct. The weight of an edge $xy$ in $G$ is the sum of the labels of the end vertices $x$ and $y$. The minimum $k$ for which the graph $G$ has an edge irregular $k$-labeling is called the edge irregularity strength of $G$, denoted by $\mathrm{\text{es}}(G)$. In this paper, we determine the exact value of edge irregularity strength of corona product of graphs with cycle.

Let $G=(V,E)$ be a graph with edge set $E$ and vertex set $V$. For a connected graph $G$, a vertex set $S$ of $G$ is said to be a geodetic set if every vertex in $G$ lies in a shortest path between any pair of vertices in $S$. If the geodetic set $S$ is dominating, then $S$ is geodetic dominating set. A vertex set $S$ of $G$ is said to be a restrained geodetic dominating set if $S$ is geodetic, dominating and the subgraph induced by $V-S$ has no isolated vertex. The minimum cardinality of such set is called restrained geodetic domination (rgd) number. In this paper, rgd number of certain classes of graphs and 2-self-centered graphs was discussed. The restrained geodetic domination is discussed in graph operations such as Cartesian product and join of graphs. Restrained geodetic domination in corona product between a general connected graph and some classes of graphs is also discussed in this paper.

Let $G$ be a graph of which $A$ be the adjacency matrix and $D$ be a diagonal matrix whose diagonal entries are the degrees of vertices of $G$. $Q=A+D$ is known as the signless Laplacian matrix of the graph $G$. For any real $\alpha \in [0,1],A_{\alpha }$ matrix is defined by the convex combination $\alpha D+(1-\alpha )A$ of $D$ and $A$. Clearly, $A_{\alpha }$ coincides with $A$ for $\alpha =0$ and coincides with $Q/2$ for $\alpha =1/2$. Thus, the $A_{\alpha }$ eigenvalues are generalizations of adjacency eigenvalues and signless Laplacian eigenvalues.

In this paper, we have obtained the $A_{\alpha }$ spectra of corona and edge corona of two graphs. We have also shown that $A_{\alpha }$ cospectral graphs and $\alpha$ equienergetic graphs can be obtained from there.

The notion of corona of two graphs was introduced by Frucht and Harary in 1970. In this paper, we generalize their definition of corona product of two graphs and introduce corona product of two signed graphs by utilizing the framework of marked graphs, which was introduced by Beineke and Harary in 1978. We study structural and spectral properties of corona product of signed graphs. Further, we define signed corona graphs by considering corona product of a fixed small signed graph with itself iteratively, and we call the small graph as the seed graph for the corresponding corona product graphs. Signed corona graphs can be employed as a signed network generative model for large growing signed networks. We study structural properties of corona graphs that include statistics of signed links, all types of signed triangles and degree distribution. Besides we analyze algebraic conflict of signed corona graphs generated by specially structured seed graphs. Finally, we show that a suitable choice of a seed graph can produce corona graphs which preserve properties of real signed networks.

The metric dimension of graphs has been extended in some types and variations such as local metric dimension and complement metric dimension of graphs. Merging two concepts is one way of developing the concept of graph theory as a branch of mathematics. These two variations motivated us to construct a new concept of metric dimension so called local complement metric dimension. We apply this new concept to some particular classes of graphs and the corona product of two graphs. We also characterize the local complement metric dimension of graphs with certain properties, namely, bipartite graphs and odd cycle graphs. Furthermore, we discover the local complement metric dimension of the corona product of two particular graphs as well as the corona product of two general graphs.

Let G be a simple undirected graph. Three new corona products of graphs based on splitting graph of G are defined. The adjacency spectra of the three new graphs based on splitting graph of G are determined. The number of spanning trees and the Kirchoff index of the new graphs are determined using their nonzero Laplacian eigenvalues.

In this paper, we introduce a variant of strong metric dimension, called the fault-tolerant strong metric dimension. A strong resolving set $S$ for $G$ is fault-tolerant if $S\backslash \{s\}$ is also a strong resolving set, for each $s$ in $S$, and the fault-tolerant strong metric dimension of $G$ is the minimum cardinality of such a set and is denoted by ${\mathrm{\text{dim}}}_{fs}(G)$.

An injective$k$-edge-coloring of a graph $G=(V,E)$ is an assignment $\omega :E\to \{1,2,\dots ,k\}$ of colors to the edges of $G$ such that any two edges $e$ and $f$ receive distinct colors if there exists an edge $g=xy$ different from $e$ and $f$ such that $e$ is incident on $x$ and $f$ is incident on $y$. The least number of colors required by any injective edge coloring of $G$ is called the injective chromatic index of $G$ and is denoted by ${\chi }_i^{\prime }(G)$. In this paper, we give tight upper bounds of the injective chromatic index of various standard graph products and operations, including the Cartesian product, lexicographic product, corona product, edge corona product, join, subdivision and Mycielskian of a graph.

One of the topics of distance in graphs is resolving set problem. This topic has many applications in science and technology namely navigation robots, chemistry structure, and computer sciences. Suppose the set $W=\{s_1,s_2,\dots ,s_k\}\subset V(G)$, the vertex representations of $x\in V(G)$ is $r_m(x|W)=\{d(x,s_1),d(x,s_2),\dots ,d(x,s_k)\}$, where $d(x,s_i)$ is the length of the shortest path of the vertex $x$ and the vertex in $W$ together with their multiplicity. The set W is called a local $m$-resolving set of graphs G if $r_m(v|W)\ne r_m(u|W)$ for $uv\in E(G)$. The local $m$-resolving set having minimum cardinality is called the local multiset basis and its cardinality is called the local multiset dimension of $G$, denoted by $m{d}_l(G)$. In our paper, we determine the establish bounds of local multiset dimension of graph resulting corona product of tree graphs.

A proper vertex coloring of a graph $G$ is said to be locally identifying (lid-coloring) if for any pair $u,v$ of adjacent vertices with distinct closed neighborhoods, the sets of colors in the closed neighborhoods of $u$ and $v$ are different. The smallest integer $k$ for which $G$ admits a lid-coloring is called the lid-chromatic number of $G$. The corona product $G\odot H$ of two graphs $G$ and $H$ is the graph obtained by taking one copy of $G$ and $|V(G)|$ copies of $H$, and then joining the $i$th vertex of $G$ to every vertex in the $i$th copy of $H$ for every $i=1,2,\dots ,|V(G)|$, where $\left|A\right|$ denotes the number of elements in the set $A$. In this paper, the lid-chromatic number of corona product of graphs has been studied.