# When do a Regular graph has an odd eigenvalue?

Merely looking at adjacency matrix of a regular graph, without explicit calculation, can we decide that graph will have an odd eigenvalue or not?

If regularity is odd, we are sure that it will be an odd e.v for graph but when regularity is not odd, what does a graph suggest by an odd e.v.?

Does having odd e.v. has something to do with structure of graph?

Any help or reference is highly appreciated. Thank you.

• What exactly do you mean by an 'odd' eigenvalue? Do you mean that its multiplicity is an odd number or that the e.v. itself is an odd number? – j4GGy May 20 '15 at 8:10
• Eigenvalue itself is an odd number. – Sry May 20 '15 at 8:21
• Sorry if this is more of a question than an answer but do you know of any implication of the adjacency matrix having integer eigenvalues (other than the regularity constant)? I ask because in general most eigenvalues are not even integers which makes your question somewhat baffling to me :) – j4GGy May 20 '15 at 8:46
• Do you know Hadamard diagonalizable graphs? They are regular and have integer eigenvalues. (further all these e.v are even). In general I don't know is there something hidden in the structure of regular integral graphs ...Definitely i would love to know that. – Sry May 20 '15 at 9:05
• Just a fun fact: If we consider the cycle graph $C_n$ on $n$ vertices, then it has regularity $2$, i.e. the e.v.s of its adjacency matrix are in $[-2,2]$. Computing the first 20 or so, it seems that $-1$ is an ev of $C_n$ if and only if $n = 3m$. Furthermore, $1$ is an ev of $C_n$ if and only if $n = 6m$. – j4GGy May 20 '15 at 10:12