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The multiple subdivision graph of a graph $G$, denoted by $S^n(G)$, is the graph obtained by inserting $n$ paths of length 2 replacing every edge of $G$. When $n=1$, $S^1(G)=S(G)$ is the subdivision graph of $G$. Let $G_1$ be a graph with $n_1$ vertices and $m_1$ edges, $G_2$ be a graph with $n_2$ vertices and $m_2$ edges. The quasi-corona SG-vertex join $G_1△G_2$ of $G_1$ and $G_2$ is the graph obtained from $S(G_1)\cup G_1$ and $n_1$ copies of $G_2$ by joining every vertex of $G_1$ to every vertex of $G_2$, and multiple SG-vertex join $G_1\odot G_2$ is the graph obtained from $S^n(G_1)\cup G_1$ and $G_2$ by joining every vertex of $G_1$ to every vertex of $G_2$. In this paper, we calculate analytic expression of characteristic polynomial of adjacency matrix of the above two types of joins of graphs for the case of $G_1$ being a regular graph. Then we obtain their adjacency spectra for the case of $G_1$ and $G_2$ being regular graphs.

The reciprocal complementary distance (RCD) matrix of a graph $G$ is defined as $\mathrm{\text{RCD}}(G)=[r{c}_{ij}]$, in which $r{c}_{ij}=\frac{1}{1+D-d_{ij}}$ if $i\ne j$ and $r{c}_{ij}=0$ if $i=j$, where $D$ is the diameter of $G$ and $d_{ij}$ is the distance between the $i$th and $j$th vertex of $G$. The $\mathrm{\text{RCD}}$-energy [$\mathrm{\text{RCDE}}(G)$] of $G$ is defined as the sum of the absolute values of the eigenvalues of RCD-matrix of $G$. Two graphs $G_1$ and $G_2$ are said to be RCD-equienergetic if $\mathrm{\text{RCDE}}(G_1)=\mathrm{\text{RCDE}}(G_2)$. In this paper, we obtain the RCD-eigenvalues and RCD-energy of the join of certain regular graphs and thus construct the non-RCD-cospectral, RCD-equienergetic graphs on $n$ vertices, for all $n\ge 9$.

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.

The metric representation of a vertex $v$ of a graph $G$ is a finite vector representing distances of $v$ with respect to vertices of some ordered subset $W\subseteq V(G)$. The set $W$ is called a minimal resolving set if no proper subset of $W$ gives distinct representations for all vertices of $V(G)$. The metric dimension of $G$ is the cardinality of the smallest (with respect to its cardinality) minimal resolving set and upper dimension is the cardinality of the largest minimal resolving set. We show the existence of graphs for which metric dimension equals upper dimension. We found an error in a result, defining the metric dimension of join of path and totally disconnected graph, of the paper by Shahida and Sunitha [On the metric dimension of join of a graph with empty graph ($O_p$), Electron. Notes Discrete Math.63 (2017) 435–445] and we give the correct form of the theorem and its proof.