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Let $G_1$ and $G_2$ be two simple connected graphs. The invariant textit{coronal} of graph is used in order to determine the $alpha$-eigenvalues of four different types of graph equations that are $G_1 circ G_2, G_1lozenge G_1$ and the other two`s are $G_1 odot G_2$ and $G_1 circleddash G_2$ which are obtained using the $R$-graph of $G_1$. As an application we construct infinitely many pairs of non-isomorphic $alpha$-Isospectral graph.
A theory of orientation on gain graphs (voltage graphs) is developed to generalize the notion of orientation on graphs and signed graphs. Using this orientation scheme, the line graph of a gain graph is studied. For a particular family of gain graphs
Bollobas and Nikiforov [J. Combin. Theory, Ser. B. 97 (2007) 859--865] conjectured the following. If $G$ is a $K_{r+1}$-free graph on at least $r+1$ vertices and $m$ edges, then $lambda^2_1(G)+lambda^2_2(G)leq frac{r-1}{r}cdot2m$, where $lambda_1(G)$
A Neumaier graph is a non-complete edge-regular graph containing a regular clique. In this paper we give some sufficient and necessary conditions for a Neumaier graph to be strongly regular. Further we show that there does not exist Neumaier graphs w
In this paper, we introduce the concepts of the plain eigenvalue, the main-plain index and the refined spectrum of graphs. We focus on the graphs with two main and two plain eigenvalues and give some characterizations of them.
Let $G$ be a graph. For a subset $X$ of $V(G)$, the switching $sigma$ of $G$ is the signed graph $G^{sigma}$ obtained from $G$ by reversing the signs of all edges between $X$ and $V(G)setminus X$. Let $A(G^{sigma})$ be the adjacency matrix of $G^{sig