Narrow Results

A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph G, denoted by $\phi \left(G\right)$, is the maximal integer k such that G has a b-coloring with k colors. In this paper, the b-chromatic numbers of the coronas of cycles, star graphs and wheel graphs with different numbers of vertices, respectively, are obtained. Also the bounds for the b-chromatic number of corona of any two graphs is discussed.

A b-coloring of a graph $G$ is a proper coloring of the vertices of $G$ such that there exist a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph $G$, denoted by $\phi (G)$, is the largest integer $k$ such that $G$ has a b-coloring with $k$ colors. The b-chromatic sum of a graph $G(V,E)$, denoted by ${\phi }^{\prime }(G)$, is introduced and it is defined as the minimum of sum of colors $c(v)$ of $v$ for any $v\in V$ in a b-coloring of $G$ using $\phi (G)$ colors. A graph $G$ is b-continuous, if it admits a b-coloring with $t$ colors, for every $t=\chi (G),\dots ,\phi (G)$. In this paper, the $b$-continuity property of corona of two cycles, corona of two star graphs and corona of two wheel graphs with unequal number of vertices is discussed. The b-continuity property of corona of any two graphs with same number of vertices is also discussed. Also, the b-continuity property of Mycielskian of complete graph, complete bipartite graph and paths are discussed. The b-chromatic sum of power graph of a path is also obtained.

Let $G$ be a $(p,q)$ graph. An injective map $f:E(G)\to \{\pm 1,\pm 2,\dots ,\pm q\}$ is said to be an edge pair sum labeling if the induced vertex function $f^{\ast }:V(G)\to Z-\{0\}$ defined by $f^{\ast }(v)={\sum }_{eϵE_v}f(e)$ is one-to-one where $E_v$ denotes the set of edges in $G$ that are incident with the vertex $v$ and $f^{\ast }(V(G))$ is either of the form $\{\pm k_1,\pm k_2,\dots ,\pm k_{\frac{p}{2}}\}$ or $\{\pm k_1,\pm k_2,\dots ,\pm k_{\frac{p-1}{2}}\}\cup \{\pm k_{\frac{p+1}{2}}\}$ accordingly as $p$ is even or odd. A graph with an edge pair sum labeling is called an edge pair sum graph. In this paper, we prove that the graphs $(P_2\times P_m)\odot K_n^c$, $(P_m\times C_3)$, book graph and $NQ(m)$ admit edge pair sum labeling.

In this paper, we define some variants of corona of graphs namely, subdivision (respectively, $R$-graph, $Q$-graph, total) neighborhood corona, $R$-graph (respectively, $Q$-graph, total) semi-edge neighborhood corona, $R$-graph (respectively, total) semi-vertex neighborhood corona of graphs constrained by vertex subsets. These corona operations generalize some existing corona operations such as subdivision ($R$-graph, $Q$-graph, total) double neighborhood corona, subdivision vertex (respectively, edge) neighborhood corona, $R$-graph vertex (respectively, edge) neighborhood corona of graphs. First, we consider a matrix in specific form and determine its spectrum. Then by using this, we derive the characteristic polynomials of the adjacency and the Laplacian matrices of the new graphs when the base graph is regular. Also, we deduce the characteristic polynomials of the adjacency and Laplacian matrices of the above mentioned particular cases from our results.

The derived graph of a simple graph $G$, denoted by ${[G]}^{†}$, is the graph having the same vertex set as $G$, in which two vertices are adjacent if and only if their distance in $G$ is two. The spectrum of derived graph of $G$ is called the second-stage spectrum of $G$. In this paper, we will determine the second-stage spectrum (Laplace and signless Laplace spectrum) of the corona of two regular graphs with diameter less than or equal to two. The energy of the derived graph is called the second-stage energy of $G$. Here, we proved that the class of graphs ${[K_{n_1}\circ K_{n_2}]}^{†}$ is integral if and only if $n_2$ is a perfect square and the second-stage energy depends only on number of vertices. Moreover, we discuss some applications like the number of spanning trees, the Kirchhoff index and Laplacian-energy-like invariant of the newly constructed graph.