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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 with exactly four distinct eigenvalues. We also determine the Neumaier graphs with smallest eigenvalue -2.
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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 with complex units, matrix properties are established. As with graphs and signed graphs, there is a relationship between the incidence matrix of a complex unit gain graph and the adjacency matrix of the line graph.
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.
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 $lambda_1(G)$ and $lambda_2(G)$ are the largest and the second largest eigenvalues of the adjacency matrix $A(G)$, respectively. In this paper, we confirm the conjecture in the case $r=2$, by using tools from doubly stochastic matrix theory, and also characterize all families of extremal graphs. Motivated by classic theorems due to ErdH{o}s and Nosal respectively, we prove that every non-bipartite graph $G$ of order $n$ and size $m$ contains a triangle, if one of the following is true: (1) $lambda_1(G)geqsqrt{m-1}$ and $G eq C_5cup (n-5)K_1$; and (2) $lambda_1(G)geq lambda_1(S(K_{lfloorfrac{n-1}{2}rfloor,lceilfrac{n-1}{2}rceil}))$ and $G eq S(K_{lfloorfrac{n-1}{2}rfloor,lceilfrac{n-1}{2}rceil})$, where $S(K_{lfloorfrac{n-1}{2}rfloor,lceilfrac{n-1}{2}rceil})$ is obtained from $K_{lfloorfrac{n-1}{2}rfloor,lceilfrac{n-1}{2}rceil}$ by subdividing an edge. Both conditions are best possible. We conclude this paper with some open problems.
In his survey Beyond graph energy: Norms of graphs and matrices (2016), Nikiforov proposed two problems concerning characterizing the graphs that attain equality in a lower bound and in a upper bound for the energy of a graph, respectively. We show that these graphs have at most two nonzero distinct absolute eigenvalues and investigate the proposed problems organizing our study according to the type of spectrum they can have. In most cases all graphs are characterized. Infinite families of graphs are given otherwise. We also show that all graphs satifying the properties required in the problems are integral, except for complete bipartite graphs $K_{p,q}$ and disconnected graphs with a connected component $K_{p,q}$, where $pq$ is not a perfect square.
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.
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