An element $i=(v,e)$ of a graph $G$ is called  an incidence of $G$, if $v\in V(G)$, $e\in E(G)$ and $v\in e$. The simultaneous coloring of vertices and incidences of a graph is coloring  the vertices and incidences of the graph properly at the same time such that any two adjacent or incident elements receive distinct colors. In this paper, we investigate the simultaneous coloring of vertices and incidences of hypercubes.

The energy E(G) of a graph G is equal to the sum of the absolute values of the eigenvalues of G. Several classes of graphs are known that satisfy the condition E(G) > n, where n is the number of vertices. We now show that the same property holds for (i) biregular graphs of degree a, b, with q quadrangles, if q £ abn/4 and 5 £ a < b £ (a - 1)2/2 ; (ii) molecular graphs with m edges and k pendent vertices, if 6 n3 - (9m + 2k)n2 + 4m3 ³ 0 ; (iii) triregular graphs of degree 1, a, b that are quadrangle–free, whose average vertex degree exceeds a, that have not more than 12n/13 pendent vertices, if 5 £  a < b £  (a - 1)2/2.

An edge labeling of a connected graph G = (V, E) is said to be local antimagic if it is a bijection f: E →{1, . . ., |E|} such that for any pair of adjacent vertices x and y, f+(x)≠ f+(y), where the induced vertex label f+(x)= ∑ f(e), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by Xla(G), is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. In this paper, the sharp lower bound of the local antimagic chromatic number of a graph with cut-vertices given by pendants is obtained. The exact value of the local antimagic chromatic number of many families of graphs with cut-vertices (possibly given by pendant edges) are also determined. Consequently, we partially answered Problem 3. 1 in [Local antimagic vertex coloring of a graph, Graphs and Combin., 33, (2017), 275--285].

Suppose that $G$ is a simple connected graph with vertex set $V(G)$ and edge set $E(G)$. A subset $S=\{s_1, s_2,\ldots , s_l \}$ of vertices of graph $G$ is called a doubly resolving set of $G$, if for any distinct vertices $u$ and $v$ in $G$ there are elements $x$ and $y$ in the set $S$ such that $d(u, x)-d(u, y)\ne d(v, x)-d(v, y)$. The minimum size of a doubly resolving set of the vertices of graph $G$ is denoted by ${\psi} (G)$. In this paper, we calculate the resolving sets of vertices with the minimum size for the line graph $L(C_n\circ{\overline{K}}_m)$ and graph $\left((C_n\circ{\overline{K}}_m)\square P_k\right)$, in which the symbols $\circ$ and $\square$ denote the Corona product and Cartesian product between two graphs, respectively. In particular, we show that if $n\geq 3$ and $m, k\geq 2$ are integers, then ${\psi}((C_n\circ{\overline{K}}_m)\square P_k )={\psi}(C_n\circ{\overline{K}}_m)+{\psi}(P_k )-1$, which gives a partial answer to the problem of characterizing graphs $G$ and $H$ satisfying the equality ${\psi}(G\square H)={\psi}(G)+{\psi}(H)-1$, which is recently posed in [K. Nie and K. Xu, The doubly metric dimension of cylinder graphs and torus graphs, Bull. Malays. Math. Sci. Soc., 46 (2023) 19 pp]. 1. IntroductionLet $G$ be a simple and connected graph with the set of vertices $V(G)$ and the set of edges $E(G)$. We denote the length of the shortest path between two vertices $u$ and $v$ in the graph $G$ by $d(u, v)$. We use $C_n$, $\overline{K_m}$ and $P_k$ to denote the cycle graph of order $n$, complement of the complete graph on $m$ vertices and the path graph of order $k$, respectively. Also, the line graph $G$ is denoted by $L(G)$, that the set of vertices of $L(G)$ are the same as the edges of the graph $G$, and two vertices are adjacent in the graph $L(G)$, if their corresponding edges in graph $G$ have a common vertex [6]. Our goal is to calculate some resolving sets depending on the line graph of the Corona product $C_{n}\circ \overline{K_{m}} $ and the Cartesian product $(C_{n}\circ \overline{K_{m}} )\square P_{k}$, so we give first some explanations about the Corona product and Cartesian product of graphs. Suppose $G$ and $H$ are two graphs with $n$ and $m$ vertices, respectively. If we consider $n$ copies of $H$ and for $i=1, 2,\cdots ,n$, all the vertices of the $i^{th}$ copy of $H$ are adjacent to the vertex $i$ of $G$, then the desired graph is called the Corona product of two graphs $G$ and $H$ and we denote it by $G\circ H$. Also, if $G$ and $H$ are two graphs, we denote the Cartesian product of these graphs by $G \square H$ or $G \times H$ and define in this way $V\left(G\square H\right)=V(G)\times V(H)$ and two vertices $(g, h)$ and $(g',h')$ in $G\square H$ are adjacent if and only if $g=g'$ and $hh'\in E(H)$ or $h=h'$ and $gg'\in E(G)$. For any ordered subset $S=\{ s_{1}, s_{2},\cdots,s_{k}\}$ of the vertices of graph $G$ and the vertex $v$ of $G$, representation the vertex $v$ with respect to the ordered set $S$ denoted by $r(v{|S)}$ and so it is $ r(v{|S)}=\left({d}\left(v, s_1\right),{d}\left(v, s_2\right),\ldots,{d}\left(v, s_k\right)\right)$. If all the vertices of the graph $G$ have distinct metric representations with respect to the ordered set $S$, then $S$ is called a resolving set of vertices of $G$. A resolving set of vertices of $G$ with the minimum size is called the metric dimension of the graph $G$ and it is represented by ${\beta }\left(G\right)$. The study of resolving set of vertices in graph theory dates back to the 1970s, and such concepts were first introduced in the articles [7,18]. The metric dimension of complete graphs, trees, paths, and the Cartesian product have been taken into consideration, also in the article [5] all graphs of order $n$ that have the metric dimension greater than or equal to $n-2$ are fully characterised. The subset $ S=\{s_{1},s_{2},\cdots,s_{l}\}$ of the vertices of graph $G$ is called a doubly resolving set for $G$, if for any distinct pair vertices $u$ and $v$ of $G$, there are the elements $x$ and $y$ of $S$, in which $d(u, x)-d(u, y)\ne d(v, x)-d(v, y)$. We denote the size of the minimum doubly resolving set in graph $G$ by ${\psi}\left(G\right) $. Concepts related to resolving sets and doubly resolving sets of graphs have been studied in the articles [4,5]. The graph $(C_5\circ \overline{K}_{3})$ and graph $L(C_{5}\circ \overline{K}_{3})$ are drawn in Figure 1. Also, the graph $(C_3\circ \overline{K}_{3})$ and graph $(C_{3}\circ \overline{K}_{3})\square P_{2}$ are drawn in Figure 2. 2. Main ResultsTheorem 2.1. If $n$ and $m$ are fixed positive integers, in which $n\geq 3$, $m\geq 2$, then the cardinality of minimum doubly resolving set of the line graph of graph $C_n\circ{\overline{K}}_m$ is $nm-n$. Theorem 2.2. If $n$, $m$ and $k$ are fixed positive integers, in which $n\geq 3$ and $m, k\geq 2$, then ${\beta }((C_n\circ{\overline{K}}_m)\square P_k )=nm-n+1.$ Theorem 2.3. If $n$, $m$ and $k$ are fixed positive integers, in which $n\geq 3$ and $m, k\geq 2$, then ${\psi }\left((C_n\circ \overline{K}_{m})\square P_{k}\right)={\psi } (C_n\circ \overline{K}_{m})+{\psi }(P_{k})-1.$ 3. ConclusionsConsidering the importance of the minimum resolving sets in graphs, in this paper, we first calculated some resolving sets of vertices with the minimum size for the line graph of graph $C_n\circ{\overline{K}}_m$ and graph $\left((C_n\circ{\overline{K}}_m)\square P_k\right)$. In particular, we show that if $n\geq 3$ and $m, k\geq 2$ are integers, then ${\psi}((C_n\circ{\overline{K}}_m)\square P_k )={\psi}(C_n\circ{\overline{K}}_m)+{\psi}(P_k )-1$, which gives a partial answer to the problem of characterizing graphs $G$ and $H$ satisfying the equality ${\psi}(G\square H)={\psi}(G)+{\psi}(H)-1$, which is recently posed in [15].

In this paper we consider the reverse backup 2-median problem with variable coordinates of vertices in Rk. Let n points be given, then in the reverse backup 2-median problem we want to change the coordinates of these points with a limited budget such that the two given points become the backup 2-median. We present a mathematical model for this problem and then solve a spatial case of this problem by a meta-heuristic method.

In this paper, first, we explain the concept of magic graphs, and then we describe the complete magic labeling of the vertices of a graph. Also, some conditions that must be met so that this labeling can be done in complete bipartite graphs are stated.

The regular graph of ideals of the commutative ring R, denoted by G reg (R), is a graph whose vertex set is the set of all non-trivial ideals ofR and two distinct vertices I and J are adjacent if and only if either I contains a J -regular element or J contains an I -regular element. In this paper, it is proved that the radius of Greg (R) equals 3. The central vertices of G reg (R) are determined, too.

Let G be a finite group and cd (G) be the set of nonlinear irreducible character degrees of G. Suppose that  (G) denotes the set of primes dividing some element of cd (G). The bipartite divisor graph for the set of character degrees which is denoted by B(G), is a bipartite graph whose vertices are the disjoint union of  (G) and cd (G), and a vertex p 2  (G) is connected to a vertex a 2 cd (G) if and only if p|a. In this paper, we investigate the structure of a group G whose graph B(G) has five vertices. Especially we show that all these groups are solvable.

We investigate the existence of a point in the plane of a unit polygon that is at rational distance from each vertex of the polygon. A negative answer is obtained in almost all cases.